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ELEMENTS   OF  ALGEBRA 


BY 

JAMES   M.   TAYLOR,  A.M.,  LL.D. 

PROFESSOR    OF    MATHEMATICS,    COLGATE    UNIVERSITY 


3>»ic 


Boston 

ALLYN    AND     BACON 

1900 


COPTKIGHT,    19  00, 
BY  JAMES  M.  TAYLOR, 


NorfajootJ  5|«ss 

J.  S.  Gushing  &  Co.  -  Berwick  &  Smith 

Norwood  Mass.  U.S.A. 


T3  ' 


PREFACE 

The  author  has  aimed  to  make  this  treatment  of  algebra 
so  simple  that  the  pupil  can  begin  the  book  to  advantage 
immediately  upon  completing  an  ordinary  course  in  arith- 
metic j  and,  at  the  same  time,  so  scientific  that  he  will 
have  nothing  to  unlearn  as  he  advances  in  the  study  of 
mathematics.  Great  care  has  been  taken  to  develop  the 
subject  logically,  yet  the  immaturity  of  the  pupil  has  been 
constantly  kept  in  mind,  and  every  legitimate  aid  has  been 
given  him.  Simplicity  has  been  attained  not  by  using 
inexact  statements  and  mechanical  methods,  but  by  avoid- 
ing many  of  the  outgrown  phrases  of  traditional  algebra, 
by  giving  demonstrations  and  explanations  in  full,  and  by 
making  fundamental  concepts  clear  and  tangible.  An 
introductory  chapter  explains  the  meaning  and  advantages 
of  the  literal  notation,  and  illustrates  the  use  of  the  equa- 
tion in  solving  arithmetic  problems.  In  Chapter  II  real 
numbers  are  first  considered,  and  are  defined  as  multiples 
of  the  quality-units,  + 1  and  —  1,  and  the  pupil  is  drilled 
in  the  use  of  particular  real  numbers  before  he  is  required 
to  represent  general  real  numbers  by  letters. 

General  principles  are  first  illustrated  by  particular  ex- 
amples, the  study  of  which  prepares  the  pupil  to  grasp 
the  meaning  of  the  formal  statement  of  the  principles,  and 


iv  PREFACE 

makes  it  less  likely  that  lie  will  memorize  without  com- 
prehending the  demonstrations  which  follow.  With  this 
arrangement  the  reproduction  of  the  demonstrations  may 
be  left  for  the  review;  but  the  pupil  should  become 
familiar  with  each  principle  and  definition  before  a  new 
one  is  considered.  When  the  demonstrations  are  not  re- 
produced, it  is  recommended  that  the  proofs  be  carefully 
read  and  discussed  in  class,  so  that  the  pupil  may  be 
fully  convinced  that  the  principles  are  true.  He  should 
then  be  required  to  state  the  authorities  for  each  step  in 
the  proof  when  the  steps  are  given. 

The  identity  and  the  equation  are  sharply  distinguished. 
Two  groups  of  principles  are  stated,  the  first  for  proving 
the  identity,  the  second  for  solving  the  equation. 

The  need  of  the  principles  of  the  equivalency  of  equa- 
tions and  systems  is  clearly  shown.  These  principles  are 
fully  illustrated  and  proved,  and  upon  them  are  based  the 
methods  of  solving  equations  and  systems  of  equations. 
In  the  chapter  on  factoring,  the  formation  of  equations 
with  given  roots  serves  as  an  introduction  to  the  converse 
problem  of  finding  the  roots  of  a  given  quadratic  or  higher 
equation,  and  to  the  method  of  making  factoring  funda- 
mental in  the  study  and  solution  of  quadratic  and  higher 
equations  and  systems. 

The  graph  is  used  to  illustrate  the  meaning  of  equations 
in  two  unknowns,  of  systems  of  equations  and  of  equivalent 
systems;  it  also  serves  to  make  clear  some  of  the  general 
properties  of  equations  in  one  unknown. 

The  theory  of  limits  is  given  as  briefly  as  is  thought  to 
be  consistent  with  clearness.     It  is  used  in  proving  the 


PREFACE  V 

laws  of  incommensurable  numbers  and  in  evaluating  ex- 
pressions which  assume  the  indeterminate  form  0/0. 

The  treatment  of  imaginary  numbers  affords  a  good 
illustration  of  the  advantages  derived  from  regarding  alge- 
braic numbers  as  arithmetic  multiples  of  quality-units. 
When  a  pupil  understands  that  the  quality-units  V— 1 
and  —  V— 1  include  the  idea  of  the  arithmetic  one  and 
that  of  oppositeness  to  each  other,  that  (V— 1)^  =  —1,  and 
(V— 1)*=  +1,  he  has  mastered  all  that  is  new  in  imagi- 
naries,  and  can  then  state  the  general  laws  for  products 
and  quotients  of  imaginary  and  real  numbers  (§§  274,  276). 
This  concept  makes  for  simplicity,  for  it  enables  us  to 
express  general  laws  which  are  true  for  real,  imaginary, 
and  complex  numbers,  and  it  clearly  separates  the  problem 
of  finding  the  arithmetical  value  of  a  result  from  that  of 
finding  its  quality.  Graphic  representations  are  used  to 
illustrate  the  meaning  and  reality  of  imaginary  and  com- 
plex numbers. 

Special  attention  is  invited  to  the  brevity  and  complete- 
ness of  the  demonstrations  of  the, principles  of  proportion, 
the  early  introduction  of  the  remainder  theorem,  the  use 
of  type-forms  in  factoring,  and  the  treatment  of  fractional 
and  irrational  equations. 

The  methods  of  working  examples  have  been  chosen  for 
their  simplicity  and  the  scope  of  their  application.  The 
problems  are  varied,  interesting,  well  graded,  and  not  so 
difficult  as  to  discourage  the  beginner.  Many  exercises 
contain  easy  examples  which,  especially  in  the  review, 
should  be  used  for  oral  work.  Suggestions  as  to  the 
method  of  attack  are  freely  given  ;   rules  are  stated  only 


Vi  PREFACE 

for  the  most  difficult  operations,  but  not  until  after  these 
have  been  illustrated  by  particular  examples. 

The  author  has  sought  to  treat  each  subject  with  suffi- 
cient fulness  to  meet  the  college  entrance  requirements, 
and  more  subjects  are  given  than  are  ordinarily  considered 
as  a  part  of  elementary  algebra. 

The  author  is  indebted  to  many  teachers  for  valuable 
suggestions,  but  especially  to  his  assistant,  Mr.  C.  D.  Kings- 
ley,  who  has  carefully  read  all  the  manuscript  and  most 
of  the  proof  sheets. 

JAMES  M.  TAYLOR. 

Colgate  University, 
June,  1900. 


CONTENTS 


CHAPTER  I 
Introduction 

SECTIONS  PAOB 

1-8.   Symbols  of  Number  and  of  Operations     ....        1 

9-18.   Equalities  ;  Identities  and  Equations        ....        8 

19.   Problems  solved  by  Equations 15 

CHAPTER   II 

Positive  and  Negative  Numbers 

20-31.    Real  Numbers  and  their  Symbols 20 

32, 33.  Principles  of  Identities 28 

CHAPTER  III 

Addition,  Subtraction,  and  Multiplication  op  Bbal 
Numbers 


34-38.  Fundamental  Laws  of  Addition        .... 

39-44.  Definition  and  Fundamental  Laws  of  Subtraction    . 

45-50.  Definition  and  Fundamental  Laws  of  Multiplication 

51-53.  Bases  and  Integral  Positive  Exponents     . 

54,  55.  Change  of  Quality  of  Expressions    .... 

56,  57.  Two  Uses  of  +  and  — .     Abbreviated  Notation 

58-60.  Coefficients.     The  Distributive  Law 


30 
33 
36 
40 
42 
43 
44 


CHAPTER  IV 

Addition  and  Subtraction  of  Integral  Literal 
Expressions 

61-64.   Addition  of  Similar  Terms 49 

65, 66.   Addition  of  Polynomials 52 

67.    Subtraction  of  One  Expression  from  Another  ...  54 

68-70.    Removal  and  Insertion  of  Signs  of  Grouping   ...  56 

vii 


viii  CONTENTS 

CHAPTER  V 
Multiplication  of  Integral  Literal  Expressions 

SECTIONS  PAGE 

71-73.  Degree  of  an  Expression.    Homogeneous  Expressions  61 

74-76.  a- 0=0.     0»  =  0.     a'«-a"  =  a"»+" 62 

77-81.  Product  of  One  Integral  Expression  by  Another  .        .  63 

82.  Multiplication  by  Detached  Coefficients        ...  69 

CHAPTER  VI 
Division  of  Integral  Literal  Expressions 


83- 

-89. 

90, 

91. 

92- 

-94. 

95- 

-97. 

98, 

99. 

Definition  and  Laws  of  Division 71 

Definition  and  Product  of  Fractions      ....  74 

«»"/«"= rt"*-".     Distributive  Law 75 

Division  of  One  Expression  by  Another        ...  77 

0/a=0.     Division  by  Detached  Coefficients         .        .  84 

CHAPTER   VII 

Linear  Equations  in  One  Unknown 

100-103.   Definitions,  Degree,  and  Root 87 

104-110.    Principles  of  Equivalence  of  Equations         ...       88 
111-114.    Solution  of  Linear  Equations 94 

CHAPTER  VIII 

Problems  solved  by  Linear  Equations  in  One 
Unknown 

115, 116.   Problems  solved  by  Equations.     Interest  Formulas      .      99 

CHAPTER   IX 

Powers,  Products,  Quotients 

117-119.  Powers  of  Powers  and  of  Products        ....  109 

120,121.  Square  of  Any  Expression Ill 

122, 123.  The  Product  (a  -f  &)  (a  -  b)  and  (x  -\-a)  {x  +  b)         .  113 

124-128.  Powers  of  a  -f  6.     Like  Powers 115 

129, 130.  The    Quotients    (a"  -  6«)  ^  (a  -  6)    and    (a'*  ±  &")  -^ 

(a  +  b) 118 

131-133.  The  Remainder  and  the  Factor  Theorem      .        .        .122 


CONTENTS 


IX 


CHAPTER  X 
Factors  of  Integral  Literal  Expressions 

SECTIONS 

134-136.  Monomials  and  Common  Factors  . 

137, 138.  Perfect  Squares 

139, 140.  Type-forms  x"^  -\-px+  q  and  ax"^  +  6x  +  c 

141.  Type-form  a"  —  ^",  where  n  is  Even 

142, 143.  Type-forms  a*'  T  6",  where  n  is  Odd 

144, 145.  Type-forms  a*  +  ha'^b^  +  &*  and  a"  +  6»,  n  Even 

146, 147.  Perfect  Cube.     Summary.     Special  Devices 

148, 149.  Formation  and  Solution  of  Equations   . 


PAGE 

125 
127 
130 
134 
136 
138 
139 
144 


CHAPTER  XI 

Highest  Common  Factor  and  Lowest  Common 
Multiple 


150-153.  H.  C.  F.  by  Factoring 

154-157.  H.  C.  F.  by  Division 

158, 159.  L.  C.  M.  by  Factoring 

160-163.  L.  C.  M.  by  H.  C.  F. 


150 
152 
158 
160 


CHAPTER   XII 

Fractions 

164-169.  Definition  and  Quality  of  Fractious     . 

170-178.  Reduction  of  Fractions  .... 

179.  Addition  and  Subtraction  of  Fractions 

180-184.  Multiplication  and  Division  of  Fractions 

185, 186.  Complex  Fractions.     Powers  of  Fractions 


163 
165 
170 
177 
181 


CHAPTER  XIII 
Fractional  Equations 
187-190.  Equivalence  of  Fractional  Equations.    Problems 


187 


CONTENTS 


CHAPTER  XIV 


Systems  of  Linear  Equations 


SECTIONS 

PAGE 

191-195. 

Equivalent  Equations  in  Two  Unknowns      . 

.     197 

196-199. 

Systems  and  Equivalent  Systems  . 

.     199 

200-203. 

Elimination  by  Substitution  .... 

.     201 

204-207. 

Elimination  by  Addition  or  Subtraction 

.     205 

208,  209. 

Systems  in  Three  or  More  Unknowns  . 

.     213 

210,211. 

Systems  of  Fractional  Equations  . 

CHAPTER   XV 
Problems  solved  by  Systems 

.     216 

212. 

A  Determinate  Problem        .... 

.     220 

CHAPTER  XVI 

Evolution.     Irrational  Numbers 

213-219.   An  nth  Root.     A  Rational  Number.     Number  of  Roots  230 

220-222.    Principal  Root.     Evolution 232 

223-225.   Root  of  a  Power,  of  a  Product,  of  a  Quotient        .         .  232 

226,227.    </«'=(</«)''  \/^«  =  V« 234 

228, 229.    Square  Root  of  Expressions 235 

230-232.    Cube  and  Higher  Roots  of  Expressions         .        .         .239 

233-237.    Square  and  Cube  Roots  of  Decimal  Numbers        .        .  244 

238-245.   Irrational  Numbers.     Their  Approximate  Values         .  249 


CHAPTER  XVII 
Surds 


246-250.  Definitions 

251,  252.  Reduction  of  Surds        .        . 

253, 254.  Addition  and  Subtraction  of  Surds 

255-257.  Product  of  Surds.     Conjugate  Surds     . 

258-260.  Division  of  Surds.     Root  of  Monomial  Surd 

261-263.  Properties  of  Quadratic  Surds 


253 
254 
255 
257 
260 
264 


CONTENTS  XI 

CHAPTER  XVIII 
Imaginary  and  Complex  Numbers 

8KCTION8  PA6I 

264-269.  Imaginary  Units.     Imaginary  Numbers        .        .        .  266 

270,271.  Laws  of  Imaginary  Numbers.     Powersoft.        .        .  269 

272, 273.  V^^a  =  i  •  y/a.     Addition  and  Subtraction          .        .  270 

274.  Product  of  Two  or  More  Quality-numbers    .        .        .  270 

275, 276.  Quotient  of  One  Quality-number  by  Another       .        .  271 

277-286.   Complex  Numbers 272 

CHAPTER  XIX 
Quadratic  Equations  in  One  Unknown 

287, 288.  Type-form.     Completing  the  Square     .         .                 .278 

289.  Solution  by  Factoring 279 

290.  Principle  of  Equivalency  of  Equations  ....  280 
291-293.  Solution  of  General  Quadratic  Equation  .  .  .  282 
294-296.  Properties  of  Roots.     Factors  of  Expressions       .        .    285 

CHAPTER  XX 
Problems 
297.     Solution  of  Problems  by  Equations      .        .        .        .291 

CHAPTER  XXI 
Irrational  Equations 

298, 299.    Definition.     Principle  of  Equivalence  .         .        .        .300 

300.  Irrational  Equations  in  Quadratic  Form       .        .        .     304 

CHAPTER   XXn 
Higher  Equations 

301.  Higher  Equations  in  Quadratic  Form    ....     306 

302.  Binomial  Equations 308 


xu 


CONTENTS 


CHAPTER  XXIII 

Systems  involving  Quadratic  and  Higher  Equations 

SECTIONS  PAGE 


303-305.  Solution  by  Factoring   . 

306.  Solution  by  Substitution 

307.  Solution  by  Addition  or  Subtraction 

308.  Systems  of  Symmetrical  Equations 
309,  310.  Solution  by  Division      . 


311 
313 

315 
318 
321 


CHAPTER   XXIV 
Inequalities 


311-314.   Definitions  and  Principles     . 
315, 316.   Proof  and  Solution  of  Inequalities 


327 
329 


CHAPTER   XXV 

Ratio  and  Proportion 

317-323.   Definitions.     Properties  of  Ratios 
324-332.    Definition  and  Properties  of  Proportions 
333, 334.   Continual  Proportion.     Mean  Proportional 


332 
334 

338 


CHAPTER   XXVI 

Theory  of  Exponents 

335-339.  Meaning  of  Fractional  and  Negative  Exponents 
340-343.   Properties  of  Exponents        .... 
344-352.   Proofs  of  Laws  of  Exponents 


341 
343 
345 


CHAPTER   XXVII 

Indeterminate  Equations  and  Systems 

353, 354.   Division  'hj  Zero.     Forms  0/0,  a/0     . 

355.    Impossible  Equations  and  Systems 
356-358.   Defective  and  Indeterminate  Systems  . 


354 
355 
356 


CONTENTS 


XlU 


CHAPTER   XXVIII 
Theory  of  Limits 

SECTIONS  PAGB 

359-361.   Variables.     Limits  of  Variables 361 

362-375.    Principles  of  Limits 362 

376-379.    Infinitesimals.     Infinites 367 

380-382.    Indeterminate  Forms 369 

383-388.  Proofs  of  tlie  Laws  of  Incommensurable  Numbei-s        .  371 

389-398.    Variation,  Direct,  Inverse,  Joint 374 

CHAPTER   XXIX 

The  Progressions 

399-405.   Arithmetic  Progression 380 

406-414.    Geometric  Progression 386 

415-419.    Harmonic  Progression 392 

CHAPTER   XXX 
Permutations  and  Combinations 


420-424.   Permutations 
426-428.    Combinations 


395 
400 


CHAPTER  XXXI 

Binomial  Theorem 

429-435.   Exponent  a  Positive  Integer 404 

436.    Exponent  Fractional  or  Negative 408 


CHAPTER  XXXII 
Logarithms 

437,438.    Definitions 412 

439-445.    Properties  of  Logarithms  to  Any  Base  ....  413 

446.   Base  Greater  than  1 416 

447-455.  Common  Logarithms 416 

456.   Exponential  Equations 422 

457-461.    Compound  Interest  and  Annuities        ....  424 


XIV 


CONTENTS 


CHAPTER  XXXIII 
Graphic  Solution  of  Equations  and  Systems 


462.  Location  of  a  Point  in  a  Plane 

463.  Graphic  Solution  of  Equations  in  x  and  y 

464.  Graphic  Solution  of  Systems  of  Equations 
465-468.  Graphic  Solution  of  Equations  in  x 


PAGB 

430 
432 
437 
441 


CHAPTER   XXXIV 

Theory  of  Equations 

469.  Horner's  Method  of  Synthetic  Division         .        .        .     444 

470-474.  /  (x)  =  0  has  w  Roots 446 

475-477.    Equal  and  Imaginary  Roots •  448 

478-480.  Relation  of  Roots  and  Coefficients         .        .         .        .452 

481, 482.  Positive,  Negative,  and  Commensurable  Roots     .        .     454 

483.  Limits  of  Roots     .        .        .        .       • .        .        .        .457 

484.  The  Roots  of   One    Equation   Multiples  of  those  of 

Another 458 

485-487.   Location  of  Real  Roots 460 


ELEMENTS   OF   ALGEBRA 


CHAPTER  I 
INTRODUCTION 

1.  Arithmetic  number.  In  Arithmetic  we  have  seen  that 
by  taking  a  group  of  ones  we  can  obtain  any  tvhole  number ; 
and  that  by  dividing  one  into  equal  parts,  and  taking  a 
group  of  these  parts,  we  can  obtain  any  fractional  number. 

Hence,  the  primary  unit  of  arithmetic  number  is  one,  1. 

A  whole  number,  or  an  integer,  is  one  or  an  aggregate 
of  ones. 

A  fractional  unit  is  one  of  the  equal  parts  of  one. 

A  fractional  number  is  a  fractional  unit,  or  any  aggregate 
of  fractional  units  which  does  not  equal  a  whole  number. 

In  writing  fractions  we  often  use  the  sign  /;  thus, 
3/2  denotes  f. 

The  numbers  defined  above  answer  the  single  question, 
'  How  many  ? '  and  are  called  arithmetic,  or  absolute,  numbers. 

Arithmetic  numbers  are  used  to  express  how  many  times  one  quan- 
tity contains  another  of  the  same  kind.  By  diminishing  indefinitely 
the  fractional  unit  we  can  obtain  a  series  of  numbers  in  which  the 
difference  between  successive  numbers  will  be  as  small  as  we  please. 

2.  A  numeral  is  any  symbol  which  is  used  to  denote  a 
particular  number,  and  which  is  never  used  to  denote  any 
other  number.  The  more  common  numerals  are  the  Arabic 
figures,  1,  2,  3,  etc.,  and  the  Roman  letters,  I,  V,  X,  etc. 

1 


2  ELEMENTS  OF  ALGEBRA 

By  the  use  of  numerals,  as  we  have  seen  in  Arithmetic,  we  can 
state  on\y  particular  problems.  To  state  and  solve  general  problems, 
and  to  investigate  the  general  properties  of  numbers,  mathematicians 
have  invented  the  literal  notation. 

3.  Letters  denoting  numbers.  —  An  important  step  in  en- 
larging the  notation  of  number  is  the  use  of  a  letter,  as 
a,  6"  X,  or  y,  to  denote  any  number  whatever  or  an  unknown 
number. 

E.g.,  just  as  heretofore  we  have  spoken  of  5  dollars,  of  8|  miles, 
etc.,  so  sometimes  we  shall  speak  of  a  dollars,  meaning  any  number 
whatever  of  dollars ;  of  x  miles,  meaning  any  number  of  miles  or  an 
unknown  number  of  miles,  etc. 

Just  as  when  we  say  the  number  4,  or  simply  4,  we  mean 
the  number  denoted  by  the  figure  4 ;  so  when  for  brevity  we 
say  the  number  a,  or  simply  a,  we  mean  the  number  denoted 
by  the  letter  a. 

The  following  simple  examples  will  illustrate  how  letters 
are  used  to  denote  any  number  whatever  in  the  statement  of 
general  arithmetic  problems. 

Ex.  1.  If  one  merchant  has  50  dollars  and  another  has  25  dollars, 
the  two  together  have  50  +  25  dollars.  If  one  merchant  has  m  dollars 
and  another  has  n  dollars,  the  two  together  have  m  +  n  dollars. 

Here  m  or  n  denotes  any  whole  or  fractional  number ;  and  m  -\-  n 
denotes  the  sum  of  these  numbers. 

Ex.  2.  If  a  drover  buys  5  horses  at  50  dollars  each,  he  pays  50  x  5 
dollars  for  the  horses.  If  a  drover  buys  y  horses  at  x  dollars  apiece, 
he  pays  x  •><  y  dollars  for  the  horses. 

Here  y  denotes  any  whole  number,  x  any  whole  or  fractional  num- 
ber, and  XX  y  their  product. 

Ex.  3.  If  60  dollars  is  divided  equally  among  5  boys,  each  boy  re- 
ceives 60-4-6  dollars.  If  x  dollars  is  divided  equally  among  n  boys, 
each  boy  receives  x -^  n  dollars. 

Ex.  4.  li  m  men  earn  n  dollars  in  one  day,  each  man  earns  «  -5-  m 
dollars  in  one  day,  and  therefore  x  men  will  earn  w  -f-  w  x  as  dollars  in 
one  day. 


INTRODUCTION  3 

In  these  examples  the  reasoning  is  the  same  whether  the 
numbers  are  denoted  by  figures  or  by  letters. 

When  letters  are  used  in  its  statement,  each  problem  is  a 
general  problem,  and  includes  an  unlimited  number  of  par- 
ticular problems. 

4.  The  following  signs,  or  symbols,  of  operation,  with 
which,  as  has  been  assumed,  the  pupil  is  already  familiar, 
are  common  to  all  branches  of  mathematics. 

The  sign  of  addition,  +,  read  'jylus/  indicates  that  the 
number  after  the  sign  is  to  be  added  to  the  number  before  it. 

E.g.,  S  -^  i  means  that  4  is  to  be  added  to  3.  a  +  6,  read  'a 
plus  6,'  means  that  the  number  denoted  by  b  is  to  be  added  to  the  num- 
ber denoted  by  a  ;  or,  more  briefly,  it  means  that  b  is  to  be  added  to  a. 

The  sign  of  subtraction,  — ,  read  ^minus,^  indicates  that 
the  number  after  the  sign  is  to  be  subtracted  from  the  num- 
ber before  it. 

E.g..,  a-\-b  —  c,  read  'a  plus  b  minus  c,'  means  that  6  is  to  be 
added  to  a,  and  then  c  subtracted  from  this  sum. 

The  sign  of  multiplication,  x ,  or  a  point  above  the  line, 
read  ^multiplied  by,  or  Unto/  indicates  that  the  number 
before  it  is  to  be  multiplied  by  the  number  after  it. 

The  sign  of  multiplication  is  usually  omitted  between 
two  letters  or  a  figure  and  a  letter. 

E.g.^  2ab,  read  '2a6,'  means  2xaxb;  7a6c,  read  '  7  a&c,' 
means  7  •  a  •  6  •  c.  The  sign  of  multiplication  cannot  be  omitted 
between  two  factors  when  both  are  denoted  by  figures ;  for  by  the 
notation  of  Arithmetic,  54  means  60  +  4,  not  6x4. 

The  sign  of  division,  -i-,  read  ^divided  by,'  or  ' by,'  indicates 
that  the  number  before  it  is  to  be  divided  by  the  number 
after  it. 

E.g.,,  a  -i-  b  X  c  -i-  d,  read  '  a  by  &  into  c  by  d,'  denotes  that  a  is  to 
be  divided  by  6,  the  result  multiplied  by  c,  and  then  this  result 
divided  by  cl. 


4  ELEMENTS   OF  ALGEBRA 

Observe  that  in  a  series  of  additions  and  subtractions,  or 
in  a  series  of  multiplications  and  divisions^  the  operations  are 
to  he  performed  from  left  to  right. 

Exercise  1. 

1.  If  a  boy  has  5  marbles  and  wins  4  more,  how  many- 
marbles  has  he  ?  If  he  has  a  marbles  and  wins  h  more, 
how  many  marbles  has  he  ? 

2.  One  part  of  25  is  7.  What  is  the  other  part  ?  One 
part  of  25  is  n.  What  is  the  other  part  ?  One  part  of  the 
number  m  is  n.     What  is  the  other  part  ? 

3.  The  difference  of  two  numbers  is  6,  and  the  smaller 
is  12.  What  is  the  greater  ?  The  difference  of  two  num- 
bers is  n,  and  the  smaller  is  x.     What  is  the  greater  ? 

4.  How  old  will  a  man  be  in  6  years,  if  his  present  age 
is  36  years  ?  How  old  will  a  man  be  in  c  years,  if  his 
present  age  is  x  years  ? 

5.  In  10  years  a  man  will  be  50  years  old.  What  is  his 
present  age  ?  In  &  years  a  man  will  be  m  years  old.  What 
is  his  present  age  ? 

6.  The  length  of  a  room  is  x  feet,  and  its  width  is  h 
feet  less  than  its  length.     What  is  its  width  ? 

7.  One  number  is  x,  and  a  second  number  is  y  times  as 
great.     What  is  the  second  number  ? 

8.  One  number,  x,  is  y  times  as  great  as  a  second  num- 
ber.    What  is  the  second  number  ? 

9.  The  number  which  contains  4  units  and  5  tens  is 
10  X  5  +  4.  Write  the  number  which  contains  x  units  and 
y  tens. 

10.  Write  a  number  containing  a?  units,  y  tens,  and  v 
hundreds. 


INTRODUCTION  6 

11.  Of  three  consecutive  whole  numbers  6  is  the  second; 
what  are  the  first  and  the  third  ?  If  the  second  is  m,  what 
are  the  first  and  the  third  ? 

12.  Of  three  consecutive  whole  numbers  7  is  the  first; 
what  are  the  second  and  the  third  ?  If  the  first  whole 
number  is  x,  what  are  the  second  and  the  third  ? 

13.  Of  three  consecutive  even  integers,  8  is  the  third; 
what  are  the  first  and  the  second  ?  If  the  third  integer  is 
m,  what  are  the  first  and  the  second  ? 

14.  If  a  goat  costs  x  dollars,  and  a  cow  costs  4  times  as 
much  as  a  goat,  and  a  horse  costs  3  times  as  much  as  a  cow, 
how  much  does  a  horse  cost  ? 

15.  In  example  14,  how  much  do  a  goat,  a  cow,  and  a 
horse  together  cost  ? 

16.  A  is  ic  years  old,  B  is  17  years  older  than  A,  and  C's 
age  equals  the  sum  of  B's  age  and  A's  age.     How  old  is  C  ? 

17.  If  m  sheep  cost  x  dollars,  and  n  cows  cost  y  dollars, 
what  would  c  sheep  and  b  cows  cost  ? 

18.  A  travelled  a  hours  at  the  rate  of  m  miles  an  hour, 
and  B  travelled  b  hours  at  the  rate  of  y  miles  an  hour.  How 
many  miles  did  A  and  B  together  travel  ? 

19.  A  rides  his  bicycle  n  yards ;  the  circumference  of 
each  wheel  is  m  feet.  How  many  revolutions  does  each 
wheel  make  in  going  this  distance  ? 

5.  A  mathematical  expression  is  any  symbol  or  combina- 
tion of  symbols  which  denotes  a  number. 

If  all  the  symbols  of  number  in  an  expression  are 
numerals,  the  expression  is  called  a  numeral  expression. 

An  expression  which  involves  one  or  more  letters  is 
called  a  literal  expression. 

The  number  denoted  by  a  numeral  expression  is  a  par- 
ticular,  or  a  fixed,  number.     For  sake  of  distinction,  the 


6  ELEMENTS   OF  ALGEBRA 

number  which  is  denoted  by  a  literal  expression  is  called  a 
general,  or  an  arbitrary^  number. 

By  the  value  of  an  expression  we  mean  the  number  de- 
noted by  it. 

E.g.,  4,  5  —  3,  and  7x5  +  4x2  are  numeral  expressions,  and  each 
denotes  a  particular,  or  fixed,  number ;  while  a,  a +4,  and  ax-\-b  —  c-i-y 
are  literal  expressions,  and  each  denotes  a  general  number. 

6.  An  axiom  is  a  truth  so  obvious  that  it  may  be  taken 
for  granted. 

Two  numbers  are  said  to  be  equal  when  they  bear  the 
same  relation  to  the  same  unit. 

E.g.,  4x3  and  6  x  2  are  equal  numbers,  since  each  is  12  times  1. 
I  and  I  are  equal  numbers,  since  each  is  6  times  \. 

The  axioms  concerning  equal  numbers,  which  are  most 
frequently  used  in  Algebra,  as  in  Arithmetic,  are  the  fol- 
lowing : 

1.  Any  number  is  equal  to  itself. 

2.  Any  number  is  equal  to  the  sum  of  all  its  j)cirts. 

3.  If  each  of  two  numbers  is  equal  to  the  same  number, 
they  are  equal  to  each  other. 

4.  If  equal  numbers  are  added  to  equal  numbers,  the  sums 
are  equal. 

5.  If  equal  numbers  are  subtracted  from  equal  numbers, 
the  remainders  are  equal. 

6.  If  equal  numbers  are  multiplied  by  equal  7iumbers,  the 
products  are  equal. 

7.  If  equal  numbers  are  divided  by  equal  numbers,  except 
zero,  the  quotients  are  equal. 

E.g.,  12  =  8  +  4,  and  12  --  4  =  (8  +  4)  --  4. 

Again,  2x0  =  5x0;  but  we  cannot  divide  by  0  and  say  that  2  =  5. 

8.  The  value  of  a  mathematical  expression  is  not  changed 
when,  for  any  number  in  it,  an  equal  number  is  substituted. 


INTRODUCTION  1 

7.  The  following  signs  of  relation  are  common  to  all 
branches  of  mathematics: 

The  sign  of  equality,  =,  read  'is  equal  to,^  is  placed  be- 
tween two  expressions  to  indicate  that  they  denote  equal 
numbers. 

The  sign  of  inequality,  >,  read  Us  greater  than/  is  placed 
between  two  expressions  to  indicate  that  the  first  denotes  a 
greater  number  than  the  second.  The  sign  <  is  read  'is 
less  thanJ 

E.g.,       4  4-  8  >  10  is  read  '  4  plus  8  is  greater  than  10 ' ; 

and  7  —  2  <  12  is  read  '  7  minus  2  is  less  than  12.' 

Observe  that  in  each  case  the  small  end  of  the  symbol  is  toward 
the  less  number. 

The  sign  ^,  read  'is  not  equal  to/  is  used  in  stating 
that  two  numbers  are  unequal,  without  indicating  which  is 
the  greater.     Thus,  a^h\?,  read  ' a  is  not  equal  to  6.' 

8.  The  signs  of  grouping  are  the  parentheses  (  ),  the 
brackets  [  ],  the  braces  \  \,  and  the  vinculum . 

Each  of  these  symbols  indicates  that  the  expression  in- 
cluded by  it  is  to  be  treated  as  a  whole. 

E.g.,  the  expression  12  —  (3  +  5)  denotes  that  the  sum  3  +  5  is  to 
be  subtracted  from  12  ;  that  is, 

12  _  (3  +  5)  =  12  -  8  =  4. 

The  expression  [32  -  (4  +  6)  -r-  6]  -=-  3  denotes  that  one-fifth  of  the 
sum  4  +  G  is  to  be  subtracted  from  32,  and  the  remainder  divided  by 

3  ;  that  is, 

[32  _(4  +  6)-  6]-  3  =[32  -  2]-  3  =  10. 

Wlien  one  sign  of  grouping  is  used  within  another,  to  avoid  ambi- 
guity different  forms  must  be  used  as  above. 

9.  Classification  of  expressions.  A  term  is  any  expression 
in  which  the  symbols  of  number  are  not  connected  by  the 
sign  -I-  or  —  ;  as  4  x  5  ^  2  or  3  a6  -h  c. 


8  ELEMENTS   OF  ALGEBRA 

Hence  the  signs  x  and  -;-  indicate  operations  within  a 
term,  and  the  parts  of  an  expression  which  are  connected 
by  the  sign  +  or  —  are  its  terms. 

E.g.,,  each  of  the  expressions  5,  a,  and  5 aj  -r-  a  is  a  term. 
The  expression  2  ax  +  3  6  -^  c  consists   of   two  terms,   2  ax   and 
Sh-^c. 

In  this  definition  of  a  term  an  expression  ivithin  a  sign  of 
grouping  must  be  considered  as  a  single  symbol  of  number. 
Hence  a  factor  or  a  divisor  in  a  term  can  itself  consist  of 
two  or  more  terms. 

E.g.,  the  expression  (a  +  &)(c  +  (?)  is  a  term  in  which  each  of 
the  factors,  a  +  b  and  c  -\-  d,  consists  of  two  terms. 

A  monomial  is  an  expression  of  one  term  ;  as  4,  6  xy,  or 

A  polynomial  is  an  expression  of  two  or  more  terms ;  as 
4  +  T  or  a  +  3  a^?/  +  7  6. 

A  polynomial  of  two  terms  is  called  a  binomial. 

A  polynomial  of  three  terms  is  called  a  trinomial. 

Observe  that  all  operations  within  each  of  two  terms 
must  be  performed  before  performing  the  operation  between 
them. 

E.g.,  the  binomial  10  - (4  +  2) (7  -  3) -^ (6  +  2)  denotes  that. 4  +  2 
is  to  be  multiplied  by  7  —  3,  this  product  divided  by  6  +  2,  and  the 
resulting  quotient  subtracted  from  10. 


Exercise  2. 

Express  in  its  simplest  form  the  number  denoted  by  each 
of  the  following  numeral  expressions : 

1.  14 +  (7 -4).  5.  18 -(6 -2)3. 

2.  18 -(12 -7).  6.  (6  +  9)^5. 

3.  (6 +  2) -(7 -3).  7.  16 -(7 -1)^3. 

4.  (3  +  8)3.  8.  22-(18-6)-^4. 


INTRODUCTION  9 

9.    12 +[4 -(5 -3)].  11.    19 -[(2 +  4) -(5 -3)]. 

10.    18 -[8 -(4 +  2)].  12.   22  -  [23  -  (7  -  4)]  -  5. 

13.  How  many  terms  in  each  of  the  expressions  found  in 
examples  1  to  12  inclusive  ? 

14.  Find  the  sum  of  5  a;  and  7  x. 

Just  as  5  +  7  =  12,  so  bx+1  x  =  V2 x. 

Reduce  each  of  the  following  expressions  to  its  simplest 
form : 

15.  2x-{-4:X.      17.    ^x-\-^x  —  (dx.      19.    \x-ir\x-\-^x. 

16.  bx-\-lx.      18.    ^x  —  ^x-\-2x.      20.    la  +  ^a  —  \a. 

10.  An  equality  is  the  statement  that  two  expressions 
denote  the  same  number.  The  expression  to  the  left  of  the 
sign  of  equality  is  called  the  first  member  of  the  equality, 
and  the  expression  to  the  right  of  this  sign  is  called  the 
second  member. 

E.g.,  (5  +  3)0  =  72  is  an  equality ;  of  which  (5  +  3)9  is  the  first 
meuiber  and  72  is  the  second  member. 

11.  Zero  is  the  number  obtained  by  subtracting  any  num- 
ber from  itself ;  that  is,  zero  is  dehned  by  the  equality 

a  -  a  =  0.  (1) 

12.  To  find  the  value  of  a  given  literal  expression  when 
each  of  its  letters  has  some  particular  value,  we  substitute 
for  each  letter  its  particular  value,  and  simplify  the  result- 
ing expression. 

Ex.  Find  the  value  of  the  expression  (x  +  y)z  -h(a  —  b),  when 
a;  =  6,  ?/  =  3,  0  =  4,  a  =  9,  6  =  2. 

Substituting,  6  for  x,  3  for  y,  4  for  z,  9  for  a,  and  2  for  b,  in  the 
given  expression,  we  obtain 

(X  +  ?/)0  -4-(a  -  6)  =  (6  +  3)  X  4  -(9  -  2)  (1) 

=  36-7.  (2) 


10  ELEMENTS   OF  ALGEBRA 

In  the  work  above  we  have  three  equalities ;  by  axiom  8,  the  first 
expression  is  equal  to  the  second  and  the  second  is  equal  to  the  third ; 
hence,  by  axiom  3,  the  first  is  equal  to  the  third. 

13.  In  working  examples  the  student  should  give  heed  to 
the  following  suggestions : 

1.  Too  much  importance  cannot  be  attached  to  neatness 
of  style  and  arrangement.  Neatness  is  in  itself  conducive 
to  accuracy. 

2.  It  should  be  clearly  brought  out  how  each  result  fol- 
lows from  the  one  before  it ;  for  this  purpose  it  will  some- 
times be  advisable  to  add  short  verbal  explanations. 

3.  Unless  the  members  are  very  short  the  signs  of  equal- 
ity in  the  steps  of  the  work  should  be  placed  one  under  the 
other. 

Exercise  3. 

Find  the  .value  of  each  of  the  following  expressions  when 
a  =  5,  6  =  3,  c  =  4,  x  =  ^'. 

1.  a-\-h.  6.  (a  -f  h)x.  11.  x-^  {a  —  c). 

2.  a—b.  7.  {a—b)c.  12.  {a-\-b)(G -\- x). 

3.  a-\-b  —  c.  8.  {a-\-b)^x.  13.  (a  —  b)  (x  —  c). 

4.  abc.  9.  (a  —  b) -^  x.  14.  [x  —  (b -{- 1)]  a. 

5.  ab  ~- c.  10.  x^(a-\~c).  15.  [a;  +  (a  —  c)] -j- a. 

16.  [3  6-(aj-a)]--c.  18.    (9-a)(2b-c)(2x-3b). 

17.  (3a-2b)-h{x-b).      19.    (3x-4.c)(3b-2  c)^(x-c). 

20.  [2  a  -  (3  6  -  2  c)] --  [(3  c  -  3  6)  (2  a  -  3  6)]. 

21.  [3ir-2(a-6)]-f-[(2ic-3&)(a-c)]. 

22.  (2x-3b)(Aa-3x)-^{3x-3c-b). 

14.  A  proof  is  a  course  of  reasoning  by  which  the  truth 
of  a  statement  is  made  clear,  or  is  established. 


INTRODUCTION  11 

15.  Identical  expressions.  Two  numeral  expressions  which 
denote  the  same  number  or  any  two  expressions  which  de- 
note equal  numbers  for  all  values  of  their  letters  are  called 
identical  expressions. 

E.g.,  the  numeral  expressions  3G  -^  4  and  13  —  4  are  identical,  for 
each  denotes  the  number  9. 

Again,  the  literal  expressions  3  a;  +  7  x  and  6  x  +  4  x  are  identical, 
for  each  denotes  the  general  number  10  x. 

To  prove  that  two  expressions  are  identical,  we  reduce  one 
to  the  form  of  the  other,  or  we  reduce  both  to  the  same  form. 

Ex.  Prove  that  the  expressions  7x  +  3x  +  2x  and  14  x  —  2  x  are 
identical. 

7x  +  3x  +  2x  denotes  12  x,  and  14  x  —  2  x  denotes  12  x  ;  hence, 
by  definition,  the  two  expressions  are  identical. 

An  equality  whose  members  are  identical  expressions  is 
called  an  identity. 

The  sign  of  identity,  =,  read  Us  identical  with,''  is  often 
used  instead  of  the  sign  =  in  writing  a  literal  identity,  i.e., 
one  whose  members  involve  one  or  more  letters. 

E.g.,  9  +  6  =  6x3,  (1) 

or  3  X  +  7  X  =  8  X  +  2  X,  (2) 

is  an  identity,  (1)  being  numeral  and  (2)  being  literal. 

Any  equality  which  involves  only  numerals  is  an  identity. 

The  sign  =  points  out  the  fact  that  equality  (2)  is  an  identity. 

The  pupil  should  now  prove  the  identities  in  Exercise  4. 

16.  Letters  denoting  unknowns.  Any  problem  involves 
one  or  more  numbers  whose  values  are  given,  and  one  or 
more  numbers  whose  values  are  to  he  found.  Numbers  given 
are  called  knowns,  numbers  to  he  found  are  called  unknoivns. 
An  unknown  is  usually  denoted  by  one  of  the  last  letters 
of  the  alphabet ;  as  x,  y,  z. 

The  following  simple  problems  illustrate  the  advantage  of 
denoting  an  unknown  by  a  letter. 

Prob.  1.  The  sum  of  two  numbers  is  80,  and  the  greater  is  3  times  the 
less.     Find  the  numbers. 


12  ELEMENTS  OF  ALGEBBA 

Let  X  =  the  less  number  ; 

then,  since  the  greater  is  three  times  the  less, 

^x  =  the  greater  number. 
Hence  their  sum  =  a;  +  3x  =  4a;. 

Therefore,  by  the  conditions  of  the  problem,  we  have 

4  a;  =  80.  (1) 

Divide  by  4,  x  =  20,  the  less  number. 

Multiply  by  3,  Zx  =  60,  the  greater  number. 

Observe  that  the  numbers  20  and  60  satisfy  the  conditions  of  the 
problem ;  that  is,  20  +  60  =  80,  and  60  =  20  x  3. 

Prob.  2.  A  farmer  bought  a  horse,  a  cow,  and  a  goat ;  the  horse  cost 
3  times  as  much  as  the  cow,  and  the  cow  4  times  as  much  as  the  goat, 
and  all  three  together  cost  255  dollars.     What  was  the  cost  of  each  ? 

Let  X  =  the  number  of  dollars  the  goat  cost ; 

then  4  ic  =  the  number  of  dollars  the  cow  cost, 

and  12  x  =  the  number  of  dollars  the  horse  cost. 

Hence  the  number  of  dollars  all  three  cost 

=  x  -}-  4  X  +  12  a;  =  17  ic. 

Therefore,  by  the  conditions  of  the  problem,  we  have 

17  X  =  255.  (2) 

Divide  by  17,  x  —  15. 

Multiply  by  4,  4  a;  =  60. 

Multiply  by  3,  12  a;  =  180. 

Hence  the  goat  cost  $  15,  the  cow  f  60,  and  the  horse  •$  180. 

17.  Equations.  Any  equality  which  is  not  an  identity  is 
called  an  equation,  as  (1)  or  (2)  in  §  16. 

A  value  of  x  in  an  equation  in  x  is  any  number  which 
when  substituted  for  x  makes  the  equation  an  identity. 

An  equation  in  one  unknown  as  x  restricts  x  to  one  value 
or  to  a  definite  number  of  values. 


INTRODUCTION  13 

E.g.,  if  in  the  equation 

2  a;  +  2  =  a;  +  8,  (1) 

we  put  6  for  x,  we  obtain  the  identity 

2x6  +  2  =  6  +  8. 

Hence  6  is  a  value  of  x  in  equation  (1);  and,  as  will  be  seen 
later,  6  is  the  only  value  of  a;  in  (1). 

An  equation,  as  (1)  or  (2)  in  §  16,  expresses  in  symbols 
the  conditions  of  a  problem ;  and  it  restricts  its  unknown  to 
such  values  as  will  satisfy  these  conditions. 

Thus  th«  equation  4  a;  =  80  restricts  x  to  the  one  value  20 ; 
and  the  equation  3  «  =  15  restricts  x  to  the  one  value  5. 

The  two  kinds  of  equalities,  equations  and  identities,  must 
be  clearly  distinguished  the  one  from  the  other. 

An  equation  states  a  condition,  and  the  values  of  the 
unknown  which  satisfy  this  condition  are  to  be  found ;  while 
an  identity  states  that  one  of  two  expressions  can  be  reduced 
to  the  other,  and  is  to  be  proved. 

Exercise  4. 
Prove  each  of  the  following  identities : 
1.  7x3x2  =  (10-3)x6.       4.     2X+   7  a;  =  (27 --3).  a;. 
2.88-4       =(7+4)x2.       5.     9a +  loa  =(  6x4)  •  a. 
3.  204-6    =(12+5)x2.       6.  106  +  8  6  =(36^2)  •  6. 

By  inspection  find  a  value  of  x  in  each  of  the  following 
equations,  and  verify  it  by  substitution : 

7.  a;-4  =  0.  10.  4a;  =  20.  13.  3x  +  l  =  10. 

8.  2a;  =  14.  11.  2a;  +  l  =  7.  14.  a;  -  1  =  6. 

9.  3a;-15  =  0.         12.  2a;+4  =  8.         15.  2a;-4  =  4. 

16.  2  a;  +  1  =  a;  +  3.  17.  2  a;  -  1  =  a;  +  2. 


14  ELEMENTS   OF  ALGEBRA 

18.  The  following  principles,  which  are  proved  in  Chap- 
ter VII,  are  used  in  finding  the  values  of  the  unknown  in 
an  equation : 

(i)  If  the  same  number  is  added  to  or  subtracted  from  both 
members  of  an  equation,  the  unknown  has  the  same  values  in 
the  derived  equation  as  in  the  given  one. 

(ii)  If  both  members  of  an  equation  are  multiplied  or  divided 
by  the  same  known  number  (except  zero),  the  unkjioivn  has  the 
same  values  in  the  derived  equation  as  in  the  given  one. 

Ex.  1.   Find  the  value  of  x  in  the  equation 

2a:  +  5  =  ll.  (1) 

Subtracting  5  from  each  member,  we  remove  all  the  known  terms 
from  the  first  member,  and  obtain 

2  X  =  6.  (2) 

Dividing  each  member  by  2,  we  obtain 

x=3.  (3) 

By  principle  (i),  x  has  the  same  value  in  (2)  as  in  (1);  and  by  (ii), 
X  has  the  same  value  in  (3)  as  in  (2). 

Hence  3  is  the  one  and  only  value  of  x  in  equation  (1). 

Ex.  2.    Find  the  value  of  x  in  the  equation 

4x-2  =  a;  +  4.  (1) 

Add  2,  4x  =  x  +  6.  (2) 

Subtract  ic,  3  a;  =  6.  (3) 

Divide  by  3,  x  =  2.  (4) 

By  principle  (i),  x  has  the  same  value  in  (2)  as  in  (1),  and  the 
same  in  (3)  as  in  (2) ;  by  (ii),  x  has  the  same  value  in  (4)  as  in  (3) . 
Hence  2  is  the  one  and  only  value  of  a;  in  (1). 

Check  :  Putting  2  for  x  in  (1),  we  obtain  the  identity 

4x2-2  =  2  +  4. 
Hence  2  is  a  value  of  x  in  (1). 


IN  TB  OD  UCTION  1 5 

Ex.  3.    Find  the  value  of  x  in  the  equation 

fx-fx  =  |.  (1) 

To  clear  (1)  of  fractions,  we  multiply  both  its  members  by  8, 
I.e.,  by  the  least  common  multiple  of  its  denominators. 

12  X  -  10  a;  =  7,  or  2  x  =  7.  (2) 

Divide  by  2,  x  =  |.  (3) 

By  (ii),  X  has  the  same  values  in  (2)  as  in  (1),  and  the  same  in 
(8)  as  in  (2);  hence  |  is  the  one  and  only  value  of  x  in  (1). 

The  foregoing  examples  illustrate  the  method  of  finding 
the  value  of  the  unknown  in  a  simple  equation. 

Exercise  5. 
Find  the  value  of  x  in  each  of  the  following  equations : 

1.  3a;-7=2aj  +  3.  11.    5.x- -  2  =  3a;  + 4. 

2.  3a;  +  4  =  a;-hl0.  12.    Tx- 9  =  17  +  2a;. 

3.  4a;  +  4  =  a;  +  7.  13.    |ic-4  =  5-Ja;. 

4.  7a;  +  5  =  a;  +  23. 

5.  8a;  =  5a;+42. 

6.  6.T  — 5  =  4a:  +  l. 

7.  18a;- 7  =  43 -7a;. 


14. 

4a;-3  =  7-^a;. 

15. 

l^-\  =  \-\^- 

16. 

i^-\  =  i-\x. 

17. 

7a;  +  21  =  45-5a;. 

18. 

^  +  f  =  H-i.'^'- 

19. 

\x  +  l=^\x  +  \. 

20. 

A^  +  it  =  A^  +  4. 

9.    19a; -11  =  15  + 6a;. 
10.   3  a; +  15  =  a; +  25. 

19.  Problems  solved  by  equations.  Read  the  problem  care- 
fully to  find  out  exactly  what  it  means;  then  state  in 
algebraic  symbols  just  what  it  says. 

To  do  this,  let  x  denote  the  unknown  number;  or,  if 
there  are  two  or  more  unknown  numbers,  let  x  or  some 
multiple  of  x  denote  one  of  them,  and  then  express  each  of 
the  others  in  terms  of  x. 


16  ELEMENTS  OF  ALGEBRA 

By  an  equation  express  the  condition  which  the  problem 
imposes  on  x. 

Then  find  the  value  of  x  in  this  equation. 

Exercise  6. 

1.  A  line  30  inches  long  is  divided  into  two  parts,  one 
of  which  is  double  the  other.     How  long  are  the  parts  ? 

Let  X  =  the  number  of  inches  in  the  second  part ; 

then  2  aj  =  the  number  of  inches  in  the  first  part. 

Hence  the  number  of  inches  in  the  two  parts  =  2  a;  +  a;  =  oic. 
Therefore,  by  the  conditions  of  the  problem,  we  have 

3x  =  30. 

Divide  by  3,  x  =  10,  number  in  second  part. 

Multiply  by  2,  2  a;  =  20,  number  in  first  part. 

2.  A,  B,  and  C  together  have  f  90.  B  has  twice  as 
much  as  A,  and  C  has  as  much  as  A  and  B  together.  How 
much  has  each  ? 

Let  X  =  the  number  of  dollars  A  has ; 

then  2x  =  the  number  of  dollars  B  has  ; 

hence  3  a;  =  the  number  of  dollars  C  has. 

.-.  ic  +  2  .r  +  3  X  =  90. 

3.  The  sum  of  the  ages  of  A  and  B  is  67  years,  and  A 
is  17  years  older  than  B.     What  is  the  age  of  each  ? 

Ans.  42  and  25  years. 

4.  Three  men,  A,  B,  and  C,  trade  in  company  and  gain 
^  600,  of  which  A  is  to  have  3  times  as  much  as  B,  and  C 
as  much  as  A  and  B  together.     What  is  the  share  of  each  ? 

Let  X  =  the  number  of  dollars  B  is  to  have,  etc. 

5.  A  farmer  bought  3  cows  for  $  180,  and  the  prices 
paid  were  as  the  numbers  1,  2,  and  3.  What  was  the  cost 
of  each  ? 


INTRODUCTION  17 

Let        X  —  the  number  of  dollars  paid  for  the  first ; 
then  2x  =  the  number  of  dollars  paid  for  the  second, 

and  3  X  =  the  number  of  dollars  paid  for  the  third. 

6.  Divide  500  into  two  parts  which  are  as  the  numbers 
1  and  4. 

7.  What  number  is  that  whose  double  exceeds  its  half 
by  27? 

8.  Divide  $  575  between  A  and  B  so  that  A  may  receive 
$  75  more  than  B. 

Let  X  =  the  number  of  dollars  B  receives  ; 

then  X  4-  75  =  the  number  of  dollars  A  receives  ; 

hence  2  x  +  76  =  675.  (1) 

9.  Divide  105  into  two  parts  whose  difference  is  45. 

10.  What  number  is  that  to  which  if  40  is  added  the 
sum  will  be  3  times  the  original  number  ? 

11.  Divide  $84  among  A,  B,  and  C,  so  that  B  shall 
have  $  13  more  than  A,  and  C  $  16  more  than  B. 

12.  Three  men,  A,  B,  and  C,  contribute  to  an  enterprise 
$  2400.  B  put  in  twice  as  much  as  A,  and  C  put  in  as  much 
as  A  and  B  together.     How  much  did  each  contribute  ? 

13.  Find  two  numbers  whose  difference  is  10,  and  one  of 
which  is  3  times  the  other. 

14.  If  two  men,  150  miles  apart,  travel  toward  each 
other,  one  at  the  rate  of  2  miles  an  hour,  and  the  other  at 
the  rate  of  3  miles  an  hour,  in  how  many  hours  will  they 
meet? 

15.  A  horse,  carriage,  and  harness  together  are  worth 
%  625.  The  horse  is  worth  8  times  as  much  as  the  harness, 
and  the  carriage  is  worth  $  125  more  than  the  harness. 
Find  the  value  of  each.  Ans.  $  400,  $  175,  and  $  50. 


18  ELEMENTS   OF  ALGEBBA 

16.  A  man  bought  a  cow,  a  sheep,  and  a  hog  for  $  80 ; 
the  cow  cost  $  32  more  than  the  sheep,  and  the  sheep  $  6 
more  than  the  hog.     Find  the  price  of  each. 

Ans.  f  50,  $  18,  $  12. 

17.  The  sum  of  $6000  was  divided  among  A,  B,  C, 
and  D;  B  received  twice  as  much  as  A,  C  as  much  as  A 
and  B  together,  and  D  as  much  as  A,  B,  and  C  together. 
How  much  did  each  receive  ? 

Ans.  $  500,  $  1000,  $  1500,  $  3000. 

18.  A  man  has  two  sons  and  one  daughter.  He  wishes 
to  divide  $  12,000  among  them  so  that  the  younger  son 
shall  have  twice  as  much  as  the  daughter,  and  the  older 
son  as  much  as  both  the  other  children.  How  much  must 
he  give  to  each  ? 

19.  Divide  90  into  five  parts  so  that  the  second  shall  be 
5  times  the  first,  the  third  shall  be  J  of  the  first  and  second, 
the  fourth  shall  be  ^  of  the  first,  second,  and  third,  and  the 
fifth  shall  be  2  times  the  sum  of  the  other  four. 

20.  A,  B,  and  C  enter  into  partnership  to  do  business. 
A  furnishes  5  times  as  much  capital  as  B,  and  C  furnishes 
\  as  much  as  A  and  B  together.  They  all  together  furnish 
$  18,900.     How  much  does  each  furnish  ? 

21.  A  gentleman,  dying,  bequeathed  his  property  of 
$21,840  as  follows:  to  his  son  2  times  as  much  as  to  his 
daughter,  and  to  his  widow  1\  times  as  much  as  to  both  his 
son  and  daughter.     What  was  the  share  of  each  ? 

22.  A  farmer  purchased  100  bushels  of  grain.  He  bought 
2  times  as  many  bushels  of  corn  as  of  oats,  and  2\  times  as 
many  bushels  of  wheat  as  of  oats  and  corn.  How  many 
bushels  of  each  kind  did  he  buy  ? 

23.  Three  candidates  for  an  office  polled  the  following 
votes  respectively :  B  received  3  times  as  many  votes  as  A, 
and  C  1^  times  as  many  as  A  and  B  together.     The  whole 


INTRODUCTION  19 

number  of  votes  was  11,000.     How  many  votes  did  each 
receive  ? 

24.  A  banker  loaned  to  each,  of  4  men  equal  sums  of 
money.  One  man  had  the  money  2  years,  another  2^  years, 
another  3^  years,  and  another  4^  years.  The  entire  interest 
money  received  was  $  275.     How  much  did  each  man  pay  ? 

Let  X  =  the  number  of  dollars  in  the  yearly  interest  on  the  sum 
loaned  to  each  man. 

25.  A  library  contains  9  times  as  many  historical  works, 
and  5  times  as  many  scientific  books,  as  works  of  fiction. 
The  historical  works  exceed  the  works  of  fiction  and  science 
by  10,500  volumes.     How  many  volumes  are  there  of  each  ? 

26.  A  drover,  being  asked  how  many  sheep  he  had, 
replied  that  if  he  had  3  times  as  many  as  he  then  had  and 
6  more,  he  would  have  150.     How  many  had  he  ? 

27.  The  expenses  of  a  manufacturer  for  5  years  were 
$  17,500.  If  they  increased  $  500  annually,  what  were  his 
expenses  each  of  the  five  years  ? 

28.  A  farmer  had  590  sheep  distributed  in  three  fields. 
In  the  first  field  there  were  25  more  than  in  the  second,  and 
in  the  third  there  were  15  more  than  in  the  first.  How 
many  sheep  were  in  each  field  ? 

29.  Of  a  herd  of  cows,  280  are  Jerseys,  and  these  are 
35%  of  the  entire  herd.     How  many  cows  in  the  herd  ? 

Let  X  =  the  number  of  cows  in  the  entire  herd  ;  then  ^^^  x  —  280. 

30.  A  town  lost  7%  of  its  inhabitants,  and  then  had  6045 
inhabitants.     What  was  its  population  before  the  loss  ? 

31.  What  number  increased  by  -J-  of  25%  of  itself  equals 
315? 

32.  The  annual  rent  of  a  house  is  $240,  and  this  is  8% 
of  its  value.    What  is  its  value  ? 


CHAPTER   II 
POSITIVE  AND  NEGATIVE  NUMBERS 

20.  Algebra  treats  of  tlie  equation,  its  nature,  the  methods 
of  solving  it,  and  some  of  its  applications. 

21.  In  each  of  the  equations  thus  far  considered,  the  un- 
known is  an  arithmetic  number.  But  in  many  equations 
the  unknown  cannot  be  an  arithmetic,  or  absolute,  number. 

E.g.,  take  the  equation 

^x  =  2x-n.  (1) 

Subtracting  2  x  from  each  member  of  (1),  we  obtain 

a;  =  0  -  5,  or  -  5.  (2) 

Hence  the  value  of  x  in  equation  (1)  is  denoted  by  the 
expression  —  5,  which  has  no  meaning  in  Arithmetic. 

If,  therefore,  such  an  equation  as  3  x  =  2  ic  —  5  is  to  be  of 
any  use,  we  must  so  enlarge  our  concept  of  number  as  to 
give  a  meaning  to  such  an  expression  as  —  5. 

To  gain  this  larger  idea  of  number  let  us  first  consider 
opposite  concrete  quantities. 

22.  Positive   and  negative,  or   opposite,  quantities.      Two 

quantities  are  said  to  be  opposites,  if,  when  combined  (or 
united  as  parts  into  one  whole),  any  amount  of  the  one 
destroys,  or  annuls,  an  equal  amount  of  the  other.  / 

Of  two  opposite  quantities,  we  call  one  positive  and  the 
other  negative. 

JS.g.f  debts  and  credits  are  opposites ;  for  when  they  are  com- 
bined, any  amount  of  debt  annuls  an  equal  amount  of  credit.  If  we 
call  credits  positive,  debts  will  be  negative. 

20 


POSITIVE  AND  NEGATIVE  NUMBERS  21 

Two  forces  acting  in  opposite  directions  are  opposites ;  for  when 
they  are  combined,  any  amount  of  the  one  annuls  an  equal  amount 
of  the  other.  If  one  of  these  forces  is  called  positive,  the  other  is 
called  negative. 

Distances  measured  or  travelled  in  opposite  directions  are  oppo- 
sites ;  for  when  they  are  combined,  any  distance  travelled  in  the  one 
direction  annuls  an  equal  distance  travelled  in  the  opposite  direction. 
If  one  distance  is  called  positive,  the  other  is  negative. 

The  sign  -f-  or  the  sign  —  is  often  written  before  the 
measure  of  a  concrete  quantity  to  denote  its  quality,  as 
positive  or  negative.  When  thus  used,  the  signs  +  and  — 
are  read  ^positive'  and  ^negative/  respectively,  and  are 
called  signs  of  quality. 

E.g.,  if  we  call  credits  positive,  +  $5  will  denote  $5  of  credit,  and 
—  $4  will  denote  .^4  of  debt.  If  +  8  inches  denotes  8  inches  to  the 
right,  —  9  inches  will  denote  9  inches  to  the  left.  If  +  S°  denotes  3° 
above  the  zero  point,  —  7°  will  denote  7°  below  that  point. 

If  4-  400  years  denotes  400  years  after  Christ,  —  300  years  will 
denote  300  years  before  Christ. 

In  this  chapter  and  the  next  we  shall  use  as  signs  of 
quality  the  small  signs  ■•"  and  ~,  which,  by  their  size  and 
position,  are  clearly  distinguished  from  the  signs  of  opera- 
tion, +  and  — . 

Exercise  7. 


I 


1.  If  credits  are  regarded  as  positive,  what  is  denoted 
by  +^  8  ?     By  "$  11  ?     By  +$  125  ?     By  -^  175  ? 

If  debts  are  regarded  as  positive,  what  does  each  of  the 
above  expressions  denote  ? 

2.  If  degrees  above  the  zero  point  are  regarded  as  posi- 
tive, what  is  denoted  by  +1°?  By  +22°?  By  "5°?  By 
-20°? 

3.  If  distances  measured  from  the  point  0  to  the  right 
are  regarded  as  positive,  what  is  denoted  by  '1  inches? 
By  +14  inches  ?     By  "13  inches  ? 


22  ELEMENTS   OF  ALGEBRA 

4.  If  distances  north  of  the  equator  are  regarded  as  posi- 
tive, what  is  denoted  by  +300  miles  ?     By  ~700  miles  ? 

State  in  symbols  each  of  the  following  in  two  ways : 

5.  $45  gain  and  $  2^  loss  is  equal  to  $  20  gain. 

+$45  +-|25  =+.$20,  gain  being  positive  ; 
<n\  -$45  ++$25  =-$20,  loss  being  positive. 

6.  $25  gain  and  $  30  loss  is  equal  to  $  5  loss. 

23.  Positive  one  and  negative  one.  Just  as  from  the  con- 
crete unit  $1  or  1°  we  gain  the  idea  of  the  unit  1,  so  from 
1  he  concrete  positive  and  negative  units  +$  1  and  ~$  1,  or 
+1°  and  "1°,  we  gain  the  idea  of  positive  one,  +1,  and  nega- 
tive one,  ~1. 

Positive  one,  +1,  and  negative  one,  ~1,  include  both  the  idea 
of  the  arithmetic  one  and  that  of  02)positeness  to  each  other. 

The  units  +1  and  ~1  being  opposites,  each  annuls  the 
other  when  added  to  it;  that  is,  +1  +  "!  =  0,  and  ~1++1  =  0. 

The  units  +1  and  ~1  are  called  quality-units. 

Of  quality-units,  "^1  is  taken  as  the  primary  unit. 

24.  Positive  and  negative  numbers.  Just  as  we  say  that 
+4  denotes  4  times  +1,  or  4  positive  units ;  so,  enlarging  the 
meaning  of  times,  we  shall  say  that  +(1)  denotes  -|  times  +1, 
or  I  a  positive  unit,  and  "(f)  denotes  f  times  ~1,  or  f  nega- 
tive units. 

Any  arithmetic  number  of  times  the  unit  +1  is  called  a 
positive  number,  as  +5.  Any  arithmetic  number  of  times 
the  unit  ~1  is  called  a  negative  number,  as  ~4  or  ~(f). 

Observe  that  the  only  new  idea  in  a  positive  or  a  negative 
number  is  that  of  the  quality-unit  +1  or  ~1. 

A  positive  number  and  a  negative  number  are  opposite 
numbers.     Thus  +5  and  ~4  are  opposite  numbers. 

A  positive  or  a  negative  number  answers  the  two  ques- 
tions, *How  many?'  and  ^Of  what  quality?'     Its  arith- 


POSITIVE  AND  NEGATIVE  NUMBERS  23 

metic,  or  absolute,  value  answers  tlie  first  question,  and  its 
quality-unit  the  second. 

E.  g. ,  the  arithmetic  value  of  +5  is  5,  and  its  quality-unit  is  +1 ;  the 
arithmetic  value  of  —(f)  is  |,  and  its  quality  is  negative. 

A  positive  or  a  negative  number  is  integral  or  fractional 
according  as  its  arithmetic  value  is  integral  or  fractional. 

E.g.-i  +(|)  and  -(|)  are  fractional  numbers. 

25.  Symbols  for  positive  and  negative  numbers.  A  figure 
(or  figures)  with  the  sign  +,  or  ~,  prefixed  denotes  a  particu- 
lar positive,  or  a  particular  negative  number.  The  figure 
denotes  the  arithmetic  value,  and  the  sign  +,  or  -,  denotes 
the  quality-unit  +1,  or  ~1. 

E.g.^  each  of  the  expressions  +3,  -7,  +8,  -5  denotes  a  particular 
positive,  or  a  particular  negative,  number. 

A  letter  with  the  small  sign  +  or  ~  prefixed  denotes  a 
general  2)ositive  or  a  general  negative  number.  The  letter 
denotes  a  general  arithmetic  number,  and  the  sign  +  or  ~, 
denotes  the  quality-unit  +1,  or  ~1. 

E.g.,  the  expression  +a  denotes  a  general  positive  number,  the 
letter  a  denoting  a  general  arithmetic  number,  and  the  small  sign  + 
the  quality-unit  +1. 

A  letter  not  preceded  by  a  small  sign  +,  or  -,  denotes  any 
number,  positive  or  negative,  integral  or  fractional. 

E.g.,  a  denotes  +2,  -3,  +7,  -9,  or  any  other  number,  positive  or 
negative  ;  so  also  does  h,  x,  y,  or  z. 

Hence,  a  letter  in  Algebra  denotes  an  algebraic  number 
except  when,  by  the  presence  of  a  small  sign  (+  or  ~)  before 
it,  it  is  restricted  to  an  arithmetic  value. 

26.  To  add  one  number  to  another  is  to  unite  the  one 
with  the  other  into  one  whole  or  aggregate. 


24  ELEMENTS  OF  ALGEBRA 

As  in  Arithmetic,  the  two  given  numbers  are  called  sum- 
mands,  and  the  result  is  called  the  sum. 

Ex.  1.   Add  +6  to  +4. 

Four  times  the  unit  +1  plus  6  times  the  same  unit  is  equal  to  4  +  6 
times  that  unit ;  that  is, 

+1  X  4  + +1  X  6  =  +1(4 +  6); 
or  +4  +  +6  =  +10. 

Ex.  2.   Add  -5  to  -7. 

Seven  times  the  unit  -1  plus  5  times  the  same  unit  equals  7  +  4 
times  that  unit ;  that  is, 

-7  4-  -5  =  -12. 

These  examples  illustrate  the  following  principle  : 

27.  To  add  one  number  to  another  of  the  same  quality,  find 
the  sum  of  their  arithmetic  values  and  jwefix  to  it  the  sign  of 
their  common  quality.     Or  stated  in  symbols, 

+a  +  +6  =  +(a  +  b),   -a-^-b  =  "(a  +  b). 

Proof  a  times  the  unit  +1,  or  ~1,  plus  h  times  the  same 
unit  is  equal  to  a  +  6  times  that  unit. 

Exercise  8. 

1.  What  is  the  arithmetic  (or  absolute)  value  and  the 
quality-unit   of   +7?     Of   -.15?     Of   "111?     Of   "a?     Of 

+(a  +  2)  ?     Of  -{a  +  &)  ? 

2.  Find  the  sum  of  +5  and  +7.  Of  +3  and  +11.  Of  "3 
and  -16.     Of  "7  and  "9.     Of  "10  and  "12.     Of  +7  and  +14. 

3.  Find  the  sum  of  +(i.)  and  +(f).  Of  +(f)  and  +(ii). 
Of  -(f)  and  -(t%).     Of  -(i)  and  "(i©. 

Find  the  value  of  +a  +  +6, 

4.  When  a  =  43,  6  =  63.  5.   When  a  =  23,  6  =  72. 
Find  the  value  of  -a  +  -&, 

6.    When  a  =  15,  6  =  121  7.   When  a  =  f ,  6  =  i 


POSITIVE  AND  NEGATIVE  NUMBERS  25 

What  is  the  value  of  m  -H  w, 

8.  When  m  = +24,  91  = +32? 

9.  When  m  =  -36,n=  "22  ? 

10.  When  m  =  +(f),  n  =  +(-|)  ? 

11.  When  m  =-(!),  n=-(f)  ? 

28.  r/ie  s?<m  of  two  opposite  numbers  which  are  equal 
arithmetically  is  zero.     Or  stated  in  symbols, 

^a-\--a  =  0.  (1) 

Proof.  Since  ~a  and  +a  are  opposite  numbers  equal  in 
arithmetic  value,  they  annul  each  other  when  added  (§  22). 

E.g.^  -2  +  +2  =  0,  +5  +  -5  =  0,  -7  +  +7  =  0,  +8  +  -8  =  0. 

Ex.  1.   Add  -5  to  +8. 

When  -5  is  added  to  +8,  the  5  negative  units  in  -5  annul  5  of  the 
8  positive  units  in  +8.     There  remain  8  —  6  positive  units  ;  that  is, 

+8 +  -5  =  +(8 -5)  =  +3. 

Ex.  2.   Add  -9  to  +4. 

AVhen  -9  is  added  to  +4,  4  of  the  9  negative  units  in  -9  annul  the 
4  positive  units  in  +4.     There  remain  9  —  4  negative  units  ;  that  is, 

+4  4.-9  =  -(9 -4)  =-6. 

These  examples  illustrate  the  following  principle  : 

29.  To  add  one  number  to  another  of  an  opposite  quality, 
find  the  difference  of  their  arithmetic  values  and  prefix  to  it  the 
quality-sign  of  the  number  which  is  arithmetically  the  greater. 
Or,  stated  in  symbols, 

^a-\--b  =  ^{a-  b),  when  a>b.  (1) 

+a  +  -b  =  -{b  -  a),  when  a  <  b.  (2) 

Proof.  AVhen  a>b  and  'b  is  added  to  +«,  the  b  nega- 
tive  units   in  "6   annul  b  of   the  a  positive  units  in  +a. 


26  ELEMENTS  OF  ALGEBRA 

There  remain  a  —  h  positive  units;   hence,  +(a  —  6)  is  the 
sum. 

When  a  <  &,  a  of  the  h  negative  units  in  ~h  annul  the  a 
positive  units  in  +a.  There  remain  h  —  a  negative  units  j 
hence,  ~(h  ~  a)  is  the  sum. 

Exercise  9, 

1.  To  make  the  sum  zero,  what  number  must  be  added 
to  +3  ?     To  -7  ?     To  -31  ?     To  +14  ?     To  +a  ?     To  'b  ? 

2.  Find  the  sum  of  +8  and  "6.  Of  +5  and  "7.  Of  "8 
and  +4.     Of  +11  and  "15.     Of  "5  and  +17. 

3.  Find  the  sum  of  "(i)  and  +(f).  Of  "(f)  and  +(^). 
Of  +(f)  and  -(A).     Of  +a)  and  -(,4). 

What  is  the  vahie  of  +a  +  -b, 

4.  When  a  =  43,  &  =  23  ?        6.    When  a  =  23,  6  =  43  ? 

5.  When  a  =  63,  &  =  43  ?        7.    When  a  =  43,  5  =  63? 
8.    Write  six  different  sums  each  of  which  denotes  zero. 

What  is  the  vahie  of  x-\-  y, 

9.    When  x=    -J,  y=    +9? 

10.  When  x  =  +14,  y  =  -19? 

11.  When  x  =  -(-J^),  y  =  -^{i)? 

12.  When  a.  =  +(i-i),  2/ =  -(!)? 

30.  The  sign  of  continuation  is  ...  or  ---,  either  of  which 
is  read,  '  and  so  on,'  or  '  and  so  on  to.' 

Thus,  1,  2,  3,  4,  ..-,  is  read,  '1,  2,  3,  4,  and  so  on'  indefi- 
nitely ;  2,  4,  6,  8,  ...  32,  is  read,  '  2,  4,  6,  8,  and  so  on  to  32.' 

The  sign  .-.  stands  for  hence  or  therefore. 

The  sign  •.•  stands  for  since  or  because. 


POSITIVE  AND  NEGATIVE  NUMBERS  27 

31.  The  integers  of  arithmetic  number  make  up  the 
series  (1). 

0      12      3      4      5      6      7      8      9-..  /j^a 

1 1 1 i 1 1 1 1 1 \ ^  ^ 

Writing  the  positive  and  the  negative  integers  in  opposite 
directions  from  zero,  we  obtain  series  (2). 

...  -4    -3    -2    -1      0    +1    +2    +3    +4  ...  (2\ 

1 i 1 1 1 1 1 \ 1 ^  ^ 

If  the  divisions  of  the  lines  in  (1)  and  (2)  be  taken  as  units  of 
length,  then  each  number  in  (1)  expresses  simply  its  distance  from  the 
zero  point ;  while  each  number  in  (2)  expresses  not  only  its  distance, 
but  also  its  direction,  from  the  zero  point,  distances  to  the  right  being 
regarded  as  positive. 

Note.  Arithmetic  numbers  are  not  positive  numbers.  An  arith- 
metic number  has  no  quality. 

If  to  any  number  in  series  (2)  we  add  +1,  we  obtain  the 
next  right-hand  number. 

E.g.,  -4  +  +l  =  -3,     -2+n  =  -l, 

and  so  on  for  the  entire  series. 

Hence,  if  we  say  that  a  number  is  increased  by  adding  to 
it  +1,  the  numbers  in  series  (2)  increase  from  left  to  right ; 
that  is, 

...,  -3<-2,  -2<-l,  -1<0,  0<+l,  +K+2.... 

We  have,  therefore,  the  following  properties  of  positive 
and  negative  numbers : 

(i)  Any  positive  number  is  greater  than  zero;  while  any 
negative  number  is  less  than  zero. 

(ii)  Of  two  positive  numbers  the  greater  has  the  greater 
arithmetic  value;  while  of  two  negative  numbers  the  greater 
has  the  less  arithmetic  value. 

E.g.,  +4>    0  by  +4,  -4  <    0  by  +4,  -7<   0  by  +7, 

+4  >  +2  by  +2,  -4  <  "2  by  +2,  -7  <  -3  by  +4. 


28  ELEMENTS  OF  ALGEBRA 

Note.  If  we  agreed  to  say  that  a  number  was  increased  by  adding 
to  it  -1,  tlien  the  numbers  in  series  (2)  would  increase  from  right  to 
left ;  positive  numbers  would  be  less  than  zero,  and  negative  numbers 
greater  than  zero.  By  common  consent,  however,  it  is  agreed  to  say 
as  above  that  a  number  is  increased  by  adding  to  it  +1,  the  primary 
unit. 

Exercise  10. 

Which  is  the  greater,  and  how  much  the  greater, 

1.  +3  or  +7  ?  4.      0  or  +1  ?  7.    "7  or     +3  ? 

2.  +2  or  -8  ?  5.0  or  "1  ?  8.    +2  or     "3  ? 

3.  -11  or  +2?  6.    -5  or  "9?  9.    '5  or  "11? 

10.  When  is  the  product  of  two  arithmetic  fractional 
numbers  greater  than  each  number  ?  Less  than  each  num- 
ber ?  Greater  than  one  and  less  than  the  other  ?  Can  the 
product  of  two  arithmetic  integral  numbers  ever  be  less 
than  either  number  ? 

11.  When  is  the  sum  of  tw^o  algebraic  numbers  greater 
than  each  number?  Less  than  each  number?  Greater 
than  one  and  less  than  the  other  ?  Is  the  sum  of  two  arith- 
metic numbers  always  greater  than  each  number  ? 

12.  Multiplying  by  an  arithmetic  fractional  number  in- 
volves what  two  operations  with  arithmetic  whole  numbers  ? 
Addition  of  algebraic  whole  numbers  involves  the  one  or  the 
other  of  what  two  operations  with  arithmetic  numbers  ? 

32.  In  proving  and  using  identities,  the  following  princi- 
ples concerning  identical  expressions  will  be  useful. 

These  principles  clearly  follow  from  the  definition  of 
identical  expressions  in  §  15  and  the  axioms  in  §  6. 

(i)  Any  expression  is  identical  with  itself. 

(ii)  If  eoAih  of  two  expressions  is  identical  with  a  third, 
they  are  identical  with  each  other. 


POSITIVE  AND  NEGATIVE  NUMBERS  29 

(iii)  If  two  identical  expressions  are  added  to  or  subtracted 
from  tivo  other  identical  expressions,  the  residting  expressions 
are  identical. 

(iv)  If  two  identical  expressions  are  multiplied  by  two 
other  identical  expressions,  the  products  are  identical. 

(v)  If  two  identical  expressions  are  divided  by  tico  other 
identical  expressions,  not  denoting  zero,  the  quotients  are 
identical. 

(vi)  If,  for  any  expression  in  an  identity,  an  identical  ex- 
pression  is  substituted,  the  resulting  equality  is  an  identity. 

33.   The  converse  of  an  identity  is  obtained  by  interchang- 
ing its  members  ;  that  is,  the  converse  oi  A  =  B  is  B=  A. 
li  A  =  B,  then,  from  definition,  B  =  A. 
Hence,  the  proof  of  an  identity  proves  its  converse. 

^.<7.,  in  proving  +a ++&  =  +  (a  +  &), 

we  prove  +(a  +  &)=+a  ++b. 


CHAPTER   III 

ADDITION,   SUBTRACTION,   AND  MULTIPLICATION  OF 
REAL  NUMBERS 

34.  The  positive  and  negative  numbers  defined  in  Chai^- 
ter  II  are  together  often  called  real  numbers. 

In  performing  any  operation  with  real  numbers,  we  must 
keep  in  mind  that  any  such  number  is  simply  an  arithmetic 
multiple  of  the  quality  unit  ^1  or  ~1,  and  that  arithmetic 
numbers  are  added,  subtracted,  multiplied,  or  divided  in 
Algebra  just  the  same  as  in  Arithmetic. 

35.  Addition.  Observe  that,  by  §  §  27  and  29,  the  addition 
of  one  real  number  to  another  is  reduced  to  the  additiortr  of 
one  arithmetic  number  to  another,  or  to  the  subtraction 
of  one  arithmetic  number  from  another. 

To  find  the  sum  of  three  or  more  numbers  we  add  the 
second  to  the  first,  to  this  sum  we  add  the  third,  and  so  on. 

Ex.  1.    +8  +  -  5  +  +6  +  -7  =  +3  +  +6  +  -7 

=  +9  +  -7  =  +2. 

Ex.  2.   -7  +  +5  +  -3  +  +9  =  -2  +  -3  +  +9 

=  -5  +  +9  =  +4. 

36.  The  two  following  laws  of  addition  are  constantly 
used  in  Arithmetic  and  Algebra : 

The  commutative  law.  The  sum  of  tivo  or  more  numbers 
is  the  same  in  whatever  order  they  are  added. 

That  is,      a  -{-b  +  c  =  b  -\-c  -\-a  =c  -}-  b  -\-a=  "'.      (A) 

30 


ADDITION  OF  REAL   NUMBERS  31 

Thus,  we  can  commute  siimmands  {change  their  order)  to 
suit  our  convenience  or  purpose. 

E.g.^  in  Arithmetic  we  write 

^  +  3  +  ^  +  2  +  ^  =  |  +  i  +  i  +  3  +  2.  (3) 

Here  by  a  change  of  order  we  can  add  the  fractions  first. 

Prove  each  of  the  two  following  particular  cases  of  {A) : 

Ex.1.     +4+-5++0=+4++6+-5.  §14 

Ex.  2.     +2  +-3  ++4  =+2  ++4  +-3  =+4  +-3  ++2. 

Proof  of  law  (A).  This  law  holds  true  for  arithmetic 
numbers,  as  is  learned  in  Arithmetic;  hence  the  total 
number  of  positive  units  in  a,  h,  c,  etc.,  will  be  the  same 
in  whatever  order  these  summands  are  written.  For  the 
same  reason  the  total  number  of  negative  units  in  a,  6,  c, 
etc.,  will  be  the  same  in  whatever  order  their  summands 
are  written. 

Hence  the  sum  will  be  the  same,  however  we  change  the 
order  of  the  summands;  for  equal  numbers  of  opposite 
units  always  annul  each  other. 

The  associative  law.  The  sum  of  three  or  more  numbers 
is  the  same  in  whatever  way  the  successive  numbers  are 
grouped. 

That  is,  a  +  6  +  c  =  a+(6+c).  {B) 

Thus  we  can  associate  successive  summands  (group  them) 
to  suit  our  convenience  or  purpose. 

Prove  each  of  the  two  following  particular  cases  of  ( JB) : 
Ex.  1.     +4  +-6  ++6  =+4  +  (-5  ++6). 
Ex.2.     -5 ++4 +-7  =-5 +  (+4 +-7). 

Proof  of  law  (B).     a -{- b +c  =  b -\- c  +  a  by(^) 

=  (6  +  c)  +  a       by  notation 
=  a-h(b-{-c)  by  (A) 

A  similar  proof  would  apply  to  any  other  case. 


32  ELEMENTS   OF  ALGEBRA 

The  rules  for  addition  in  Arithmetic  are  based  on  the  commutative 
and  associative  laws  just  given. 

E.g.,  to  add  45  and  23,  we  have 

45  +  23  =  40  +  5  +  20  +  3  by  notation 

=  40  +  20  +  5  +  3  ^J  (A) 

=  (40 +  20) +  (5 +  3)  by  (^) 
=  60  +  8  =  68. 

Writing  one  number  under  the  other  and  then  grouping  the  vertical 
columns,  as  we  do  in  Arithmetic,  is  but  a  convenient  way  of  applying 
laws  (A)  and  (5). 

37.  Since,  by  the  laws  of  addition  in  §  36,  we  can  change 
the  order  of  summands  and  group  them  to  suit  our  purpose, 
we  have  the  following  rule  for  adding  three  or  more  num- 
bers, some  of  which  are  positive  and  some  negative : 

Add  all  the  numbers  of  one  quality,  then  add  all  the  numbers 
of  the  opposite  quality,  then  add  the  two  resulting  sums.    • 

Ex.     -5 ++9 +-11 ++6  =-5 +-11 ++9 ++6  by  (^) 

=  -16++15=-l  by  (J?) 

'  In  practice,  the  rearrangement  and  regrouping  of  the  summands 
should  be  done  mentally  and  simultaneously. 

Exercise  11. 
By  §  37  find  the  value  of  each  of  the  following  sums: 

1.  +19 +-7 ++5.  5.    +4 +-5 ++6 +-8 ++7. 

2.  -12 ++9 +-4.  6.    -9 ++6 +-11 ++12 +-4. 

3.  -22 ++5 ++7.  7.    +15 +-9 ++7 +-8 ++11. 

4.  +42 +-9 +-3.  8.    -(|)++(2)+-(|)++(|). 

Find  the  value  of  x  -\-  y  -\-  z  -{-  v. 
9.   When  a;  ="25,  2/ ="^32,  2  =-45,  v  =+28. 
10.    When  a;  =+94,  2/ ="75,  ^=+82,  v  =-Qb. 


SUBTRACTION   OF  REAL  NUMBERS  33 

38.  From  the  definition  of  zero  it  follows  that 

a  +  0  =  a. 

That  is,  any  number  plas  zero  equals  the  number  itself. 

E.g.,  7+0  =  7,  8  +  0  =  8. 

Also,  9+(2-2)=9,  6+(5-5)  =  6. 

39.  Subtraction  is  the  inverse  of  addition.  Given  a  sum 
and  one  of  its  two  parts,  subtraction  is  the  o^^eration  of 
finding  the  other  part. 

As  in  Arithmetic,  the  given  sum  is  called  the  minuend, 
the  given  part  the  subtrahend,  and  the  required  part  the 
remainder. 

Hence,  to  subtract  any  subtrahend  from  any  minuend  is 
to  find  a  third  number,  the  remainder,  which  added  to  the 
subtrahend  gives  the  minuend. 

Ex.    +9  =  +9  +  (+5  +  -5)  §§  28,  38 

=  (+9+ +5) +  -6.  §36 

Hence, 
+9  ++5  is  the  number  which  must  be  added  to  -5  to  obtain  +9  ; 

that  is,  +9 --5  =  +9 ++5. 

Here  the  remainder  +9  ++5  is  obtamed  by  adding  to  the  minuend 
+9,  the  subtrahend  -5  with  its  quality  changed. 
This  example  illustrates  the  following  rule  : 

40.  To  subtract  one  real  number  from  another,  add  to  the 

minuend  the  subtrahend  with  its  quality  changed  from  +  to  ~, 
or  from  ~  to  '^. 

That  is,  Jtf  -  +a  =  1/  -f  "a,  (1) 

M--a  =  M+^a,  (2) 

when  M  is  any  real  number. 


34  ELEMENTS   OF  ALGEBEA 

Proof.  If  to  the  second  member  of  (1)  we  add  the  sub- 
trahend, '^a,  we  obtain  the  minuend  M;  that  is, 

(M-^-a)-^+a  =  M+(-a-^+a)  by  (5) 

=  if  §§  28,  38 

also,  (M-^+a)  +-a  =  M+  (+a  +"a)  =  M. 

Hence,  by  §  39,  the  second  member  of  (1)  or  (2)  is  a 
remainder. 

Ex.1.     -4-+7  =-4+-7  =-11. 
Ex.2.     -5 --8  =-5 ++8  =+3. 

Thus,  subtracting  any  real  number  gives  the  same  result 
as  adding  its  arithmetically  equal  opposite  number. 

E.g.,  subtracting  §200  credit  from  an  estate  is  equivalent  to  adding 
$200  debt;  and  subtracting  $300  debt  is  equivalent  to  adding  .^300 
credit. 

Subtracting  $100  income  is  equivalent  to  adding  $100  expenditure. 

Exercise  12. 
Perform  each  of  the  following  indicated  subtractions : 

1.  +19 -+7.  4.    +6-+7.  7.    -20 --25. 

2.  -23 -+12.  5.    +12 -+20.  8.    "68 --98. 

3.  -16 --30.  6.    -214 -+25.  9.    -118--120. 

What  is  the  value  of  a  —  b, 

10.  When  a  =+5,  6  =+4?       12.    When  a  ="4,  5  ="7? 

11.  When  a  =+7,  6  =+9?        13.    When  a  ="14,  5  =-11? 

14.  From  +4 +-8 ++9 +-3  subtract  +7 +"2 ++9 +-8. 

15.  From  -10 +-7 ++15 +"3  subtract  +7 +"11 +"17. 

41.  When  a  monomial  or  the  first  term  of  a  polynomial 
is  preceded  by  the  sign  of  operation  +  or  — ,  zero  is  to  be 
understood  before  this  sign  of  operation. 


SUBTRACTION  OF  REAL  NUMBERS  35 

Thus,         —  +a  =  0  —  +a  =  ~a,  —~a  =  0  —  ~a=  '^a. 
Again,        _  +5  +  +7  =  0  -  +5  +  ^7  =  "5  +  +7. 

42.  Successive  subtractions  or  successive  additions  and  sub- 
tractions can  be  performed  from  left  to  right,  one  at  a  time 
in  succession. 

Ex.1.     +8--3-+2--6=+ll-+2--6 
=+9 --6  =+15. 

We  can,  however,  express  each  term  to  be  subtracted,  as 
a  term  to  be  added,  and  then  apply  the  principle  in  §  37,  for 
finding  the  sum  of  three  or  more  numbers. 

Ex.2.     +8 --3 -+2 --6  =+8 ++3 +-2 ++6  (1) 

=+17 +-2  =+15. 

43.  Commutative  law  of  subtraction.  Since  each  term  to 
be  subtracted  can  be  expressed  as  a  term  to  be  added,  the 
commutative  law  holds  for  subtraction  as  well  as  for  addi- 
tion, provided  the  sign  of  operation  -[-  or  —  before  each  term 
is  transferred  with  the  term  itself. 

E.g.,  +7_-8+-9-+4=--8++7-+4+-9 

=  _+4_-8+-9++7. 

Exercise  13. 
Find  the  value  of  each  of  the  following  expressions : 
1.   +6-f-2-+3.      2.    -14-+9-f"4.      3.    +32 +"5 -"16. 
4.    +6--24-+3.      5.    +4--2-+3+-2-+5++3--6. 

6.  +25  -+14  +-10  ++14  --5  -+18  ++16  +"18. 

7.  -35 +-5 --32 ++24 --14 +-28 --8. 

44.  From  the  definition  of  zero  it  follows  that 

a-0  =  a. 
That  is,  any  number  minus  zero  equals  the  number  itself 


36  ELEMENTS   OF  ALGEBRA 

45.  Multiplication.  As  in  Arithmetic,  the  number  multi- 
plied is  called  the  multiplicand,  the  number  which  multiplies 
is  called  the  multiplier,  and  the  result  the  product. 

Ill  Arithmetic  the  product  9  x  3  is  obtained  by  taking  the  multipli- 
cand 9  three  times,  and  the  multiplier  3  is  obtained  by  taking  the  pri- 
mary unit  1  three  times. 

The  product  9  x  |-  is  obtained  by  dividing  the  multiplicand  9  by 
3,  and  multiplying  the  result  by  2,  and  the  multiplier  |  is  obtained  by 
dividing  the  primary  unit  1  by  3,  and  multiplying  the  result  by  2. 

In  each  case  we  obtain  the  product  by  doing  to  the  multiplicand 
just  what  is  done  to  the  primary  unit  to  obtain  the  multiplier. 

Hence,  we  define  multiplication  as  follows  : 

To  multiply  one  number  by  another  is  to  do  to  the  multi- 
plicand just  what  is  done  to  the  primary  unit  to  obtain  the 
multiplier. 

The  multiplicand  and  the  multiplier  together  are  called 
the  factors  of  the  product. 

46.  Multiplier  any  arithmetic  number.     Let  a  and  b  be  any 

two  arithmetic  numbers ;  then  b  times  a  units  of  any  Mnd  is 
equal  to  ab  units  of  that  Mnd;  that  is, 

^axb  =  +{ab),  (1) 

and  -a  xb  =  -(ab).  (2) 

E.g.,        +4x5  =  +20,  -7x4  =  -28,  -(f)  x  8  =  -12. 

47.  Multiplier  any  positive  or  any  negative  number. 

To  obtain  ^b  from  the  primary  unit  +1  we  take  that  unit 
b  times ;  hence,  by  definition,  to  multiply  any  number  by  +6 
we  take  that  number  b  times; 

that  is,  +a  X  +6  =  +a  X  6  =  +(ab),  (X) 

and  -a  X  +6  =  -a  X  5  =  ~(ab).  (2) 

Hence,  to  multiply  any  number  by  +1  is  to  take  that 
number  once ;  that  is,  "*"a  x  "^1  =  +a ;  ~a  x  ■*"!  =  "a. 


MULTIPLICATION  OF  REAL  NUMBERS  37 

To  obtain  ~b  from  the  primary  unit  +1,  we  change  the 
quality  of  that  unit  and  multiply  the  result  by  6 ;  hence,  to 
multiply  any  number  by  ~b,  we  change  the  quality  of  that 
number  and  multiply  the  result  by  b  ; 

that  is,  +a  X  ~6  =  ~a  X  6  =  ~{ab)j  (3) 

and  ~a  X  "6  =  +a  X  6  =  ^(a6).  (4) 

Hence,  to  multiply  any  number  by  ~1  is  to  change  the 
quality  of  that  number;  that  is,  "^a  x  ~1  =  ~a;  ~a  x  ~1  =  ^a. 

From  identities  (1)  and  (4)  we  have  the  law, 

Tlie  product  of  two  real  numbers  like  in  quality  is  positive. 

From  identities  (2)  and  (3)  we  have  the  law, 

The  product  of  two  numbers  opposite  in  quality  is  negative. 

These  two  laws  together  are  called  the  law  of  quality  of 

products. 

From  identities  (1),  (2),  (3),  (4),  it  follows  that 

The  arithmetic  value  of  the  product  of  two  real  numbers  is 

the  product  of  their  arithmetic  values. 

E.g.,  +5  X  +7  =  +35,  -6  x  "8  =  +48. 

+4  X  -9  =  -36,  -7  X  +8  =  -56. 

Exercise  14. 

Find  the  value  of  each  of  the  following  numeral  ex- 
pressions : 

1.  +2  X  +4.      3.     +9  X  -8.      5.  -21  X  +3.     7.  +22  x  "6. 

2.  -2  X  -7.      4.  -11  X  -1.      6.  +31  X  -1.      8.  -32  x  "4. 
9.  +10x-3  +  -8x+2.       11.  -6x-5+-^8x-4-+12x-5. 

10.  +14x-2  +  -6x-5.       12.  -9x+2  +  +16x-4--14x+3. 


88  ELEMENTS   OF  ALGEBRA 

When  a  =  +2,  6  =  "4,  m  =  "3,  n  =  +9,  x  =  +6,  find  the 
value  of  each  of  the  following  literal  expressions : 

13.  ab  -f-  mx.  16.    ax  —  nb.  19.    (a  +  b)  (n  -\-  m). 

14.  ax  +  bm.  17.    (a  —  b)  x.  20.    (a  —  b)(n  —  m). 

15.  am—  bx,  18.    (m  —  n)b.         21.    (6 —a?)  (771— n). 

48.   Continued  products.     By  §  47,  we  obtain 

+a  X  "^6  X  +c  =  +(a?>)  x  +c  =  -^(abc). 

+a  X'^b  x~c  =  +{ab)  x~c  =  -(abc). 

-^a  x~b  x~c  =  ~(ab)  x~c  =  +(abc). 

~a  x~b  x~c  =  +(ab)  x^c  =  ~(abc). 

From  these  and  similar  identities  we  have  the  following 
laws  which  are  more  general  than  those  in  §  47 : 

Aj^roduct  which  coyitains  an  odd  number  of  negative  factors 
is  negative;  any  other  product  is  positive. 

The  arithmetic  value  of  a  product  is  the  product  of  the  arith- 
metic values  of  its  factors. 

Ex.     Find  the  value  of  +3  x  "2  x  +4  x  -6  x  -5. 

The  product  is  negative,  since  there  is  an  odd  number,  3,  of  nega- 
tive factors ;  its  arithmetic  value  is  3  x  2  x  4  x  6  x  5,  or  720. 

Hence,  +3  x  -2  x  +4  x  "6  x  -  5  =  "720. 

Exercise  15. 

When  a  =  -2,  &  =  +4,  c=-6,  x  =  -S,  y  =  ~5,  find  the 
value  of  each  of  the  following  literal  expressions: 


1.  abc. 

4. 

(6  +  a)  ex. 

7.  axy  — be. 

2.  abxy. 

5. 

(x  —  y)  abc. 

8.  xy  —  abc. 

3.  abcxy. 

6. 

(b  4-  c)  axy. 

9.  x-\-  obey. 

MULTIPLICATION  OF  MEAL  NUMBERS  39 

10.  Prove  +lx-lx-lx-l  =  -l;  -lx-lx-lx-l  =  +l. 

11.  Prove  +a  X  "6  X  ~c  =  (+1  x  "1  X  ~1)  (ahc). 

12.  Prove  "a  x  "^6  X  ~c  x  ~x  =  {-l  x  +1  X  ~1  X  ~1)  (obex). 

Examples  11  and  12  illustrate  that  the  product  of  two  or  more  num- 
bers is  equal  to  the  product  of  their  quality-units  multiplied  by  the 
product  of  their  arithmetic  values. 

49.  The  two  following  laws  of  multiplication  are  con- 
stantly used  in  Arithmetic  and  Algebra : 

The  commutative  law.  Tlie  product  of  two  or  more  num- 
bers is  the  same  in  vjhatever  order  the  factors  are  multiplied. 

That  is,  abc  =  acb  =  cba="'.  (A') 

Prove  each  of  the  two  following  particular  cases  of  (A') : 
Ex.  1.  +2  X  -3  X  +4  X  -5  =  -5  X  -3  X  +2  X  +4. 
Ex.  2.    -3  X  +7  X  -2  X  -1  =  -2  X  +7  X  -1  X  -3. 

Proof  In  Arithmetic  we  have  learned  that  this  law 
holds  true  for  arithmetic  numbers.  Hence,  by  §  48,  the 
arithmetic  value  of  a  product  of  real  numbers  is  the  same 
in  whatever  order  the  factors  are  multiplied. 

From  the  law  of  quality,  in  §  48,  it  follows  that  the  quality 
of  a  product  of  real  numbers  will  be  the  same  in  whatever 
order  the  factors  are  multiplied. 

Hence,  a  change  of  order  of  factors  affects  neither  the 
arithmetic  value  nor  the  quality  of  their  product. 

The  associative  law.  Tlie  product  of  three  or  more  num- 
bers is  the  same  in  whatever  way  the  successive  factors  are 
grouped. 

That  is,  abc  =  a(bc).  (B) 

Prove  each  of  the  following  particular  cases  of  {B') : 

Ex.  1.     -3  X  +4  X  -2  =  -3  X  (+4  x  "2). 

Ex.  2.     +5  X  -0  x  -1  X  +2  =  +5  X  ("6  x  -1  X  +2). 


40  ELEMENTS   OF  ALGEBUA 

Proof,  abc  =  hca  by  (A') 

=\bc)a  by  notation 

=  a(bc)  by  (A') 

Exercise  16. 

By  using  the  commutative  and  associative  laws,  find  in 
the  simplest  way  the  value  of  each  of  the  following  ex- 
pressions : 

1.  +33  X  -21  X  -4.  4.    +144  x  "3  x  -16|. 

2.  -123  X -33^  X +3.  5.    "371  x  "7  x  "4. 

3.  +142  X -121  X -8.  6.    -333i  X -5  X -7  X +3. 

50.  A  x>ii'oduct  of  two  or  more  factors  is  multiplied  by  a 
number  if  any  one  of  the  factors  is  multiplied  by  that  number. 

Proof  (ab)  x  c  =  (ac)b  =  a(bc).  §  49 

51.  Powers.  A  product  of  two  or  more  equal  factors  is 
called  a  power.  Any  number  also  is  often  called  the  first 
power  of  itself. 

E.g.j  the  product  aa  is  called  the  second  power  of  a. 

The  product  bbb  is  called  the  third  power  of  b. 

aa  is  written  a^ ;  aaa  is  written  a^ ; 

ooa  •••  to  n  factors,  written  a%  is  read  'the  nth.  power  of  a.' 

a^  is  often  read  ^the  square  of  a/  and  a^  'the  cube  of  a.' 

In  a*",  a  is  called  a  base.  Thus  in  3^*,  3  is  a  base ;  in  a^, 
a;  is  a  base ;  in  y'",  ?/  is  a  base. 

52.  A  positive  integral  exponent  is  a  whole  number  which 
(written  to  the  right  and  a  little  above  a  base)  indicates 
how  many  times  the  base  is  used  as  a  factor,  as  3  in  a^, 
or  n  in  a". 


MULTIPLICATION  OF  REAL  NUMBERS  41 

To  avoid  ambiguity,  a  base  which  is  not  denoted  by  a 
single  symbol  must  be  enclosed  within  parentheses : 

E.g.,    (-3)2  =  -3  X  -3  =  +9,  while  -3^  =  -1  (3  x  3)  =  -9. 

Again,  (4  x  5)2  =  (4  x  5)  (4  x  6)  =  20  x  20  =  400, 

while  4  X  52  =  4  X  (5  X  5)  =  4  X  25  =  100. 

The  meaning  of  fractional  and  negative  exponents  will  be  deter- 
mined in  a  later  chapter. 

A  power  is  said  to  be  odd  or  even  according  as  its  expo- 
nent is  odd  or  even. 

53.  Quality  of  a  power.  An  odd  power  of  a  negative  base 
is  the  only  power  which  involves  an  odd  number  of  negative 
factors ;  hence,  by  the  law  of  quality  in  §  48  it  follows  that 

An  odd  power  of  a  negative  base  is  negative,  and  an  even 
power  positive ;  any  power  of  a  positive  base  is  positive. 

E.g.,  any  power  of  +1  is  +1 ;  any  even  power  of  -1  is  +1  ;  any  odd 
power  of  -1  is  '1. 

Exercise  17. 

1.  What  number  is  the  base  and  what  the  exponent  in 
-3^?  In  {Sy?  In  3xy"?  In  (Sxyy?  In  (a +  6)"? 
In  a  +  b^?     In  (x")*  ? 

Find  the  value  of  each  of  the  following  expressions : 

2.  -3\  3.    (-3)*.  4.    17-32.  5.    (17-3)2. 

Express  each  of  the  following  products  by  a  base  and 
exponent : 

6.  (xy)  (xy)  (xy)  •••  to  8  factors. 

7.  (a  +  6)  (a  +  &)(«  +  &)•••  to  12  factors. 

Express  in  symbols : 

8.  The  sum  of  the  cubes  of  x  and  y. 

9.  The  cube  of  the  sum  of  x  and  y. 


42  ELEMENTS   OF  ALGEBRA 

10.  The  sum  of  the  squares  of  a,  b,  and  c. 

11.  The  square  of  the  sum  of  a,  b,  and  c. 

54.  If,  in  any  one  of  the  identities  in  §  48,  the  quality  of 
one  factor  is  changed,  the  quality  of  the  product  is  changed, 
but  its  arithmetic  value  remains  the  same. 

This  illustrates  the  following  principle : 

TJie  quality  of  any  product  is  changed  by  changing  the  qual- 
ity of  one,  or  of  any  odd  7iumber,  of  its  factors. 

Proof  By  changing  the  quality  of  an  odd  number  of 
factors,  the  number  of  negative  factors  in  the  product  is 
changed  from  odd  to  even,  or  from  even  to  odd ;  hence,  by 
§  48,  the  quality  of  the  product  is  changed. 

Note.  When  for  brevity  we  speak  of  the  quality  of  an  expression, 
we  mean,  of  course,  the  quality  of  the  number  which  the  expression 
denotes. 

55.  The  quality  of  an  expression  is  changed  by  changing  the 
quality  of  each  of  its  terms. 

Proof  Changing  the  quality  of  a  term  does  not  affect  its 
arithmetic  value.  Hence,  changing  the  quality  of  each  term 
of  an  expression  will  simply  change  a  positive  sum  into  an 
arithmetically  equal  negative  sum,  or  vice  versa. 

This  principle  is  illustrated  by  the  fact  that  if  in  a  business  account 
we  change  debts  into  credits,  and  credits  into  debts,  the  balance  will 
not  be  changed  in  amount,  but  it  will  be  changed  from  credits  to  debts, 
or  from  debts  to  credits. 

Ex.  1.  Change  in  four  ways  the  quality  of  -4  x  +3  x  -2.  Of 
-3x-6x-7.     Of  +ax+bx-c.    Of  +ax-bx-c.    Of  +xx+yx-2. 

Ex.  2.  Change  in  two  ways  the  quality  of  -4  x  +3  -  -2  x  +7. 
Of  +a  X  -b  +  -c  X  +x.    Of  -a  X  -X  —  +6  X  +c. 


MULTIPLICATION  OF  BEAL  NUMBERS  43 

56.  Two  uses  of  the  signs  +  and  -.  Hereafter  the  larger 
signs  +  and  —  will  be  used,  not  only  as  signs  of  operation, 
but  also  with  numerals  as  signs  of  quality. 

To  avoid  ambiguity,  parentheses  will  be  used  when  needed. 

Thus,  in  the  expression, 

(+4)-(+7)  +  (-3)-(-4), 

each   sign  within  parentheses  denotes    quality,   and   each    without 
denotes  an  operation. 

Again,  (- 3)ax -(+ 4)6y +  (- 5)c2;  = -3  ax  - +4  6y +-5c5r. 

A  letter  with  the  small  sign  +  or  "  will  continue  to  be 
used  to  denote  a  general  positive  or  a  general  negative 
number. 

57.  Abbreviated  notation.  The  sign  —  is  never  omitted. 
But,  for  the  sake  of  brevity,  the  sign  -f  has  been  omitted, 
and  is  to  be  understood  in  the  two  following  cases : 

(i)  When  no  sign  is  written  before  a  monomial  or  before 
the  first  term  of  a  polynomial,  the  sign  -f  is  to  be  under- 
stood. 

(ii)  When  only  one  sign  is  written  between  two  successive 
terms  of  a  polynomial,  the  sign  +  is  to  be  understood  either 
as  a  sign  of  operation  or  as  a  sign  of  quality. 

E.g.^  2  denotes  +  2,  3  a  denotes  +  3  a,  and  a  denotes  +  1  a. 

Again,  6  —  5  denotes  the  difference  (+6)  — (+5)  or  the  sum 
(+  6)  +  (—  5)  ;  in  each  case  the  sign  +  is  understood  between  6  and  5  ; 
in  the  first  case  as  a  sign  of  quality^  and  in  the  second  case  as  a  sign 
of  operation. 

Since  (+6)  -  (+5)  =  (+6)  +  (-5), 

6  —  5  denotes  the  same  number  whether  it  is  regarded  as  ex- 
pressing the  difference  (+6)  — (+5)  or  the  sum  (+6) +  (—5). 
Again, 

7-5-f8  =  (4-7)-(+5)4-(-l-8),  or  (+7)  +  (-5)  +  (+8), 


44  ELEMENTS   OF  ALGEBRA 

according  as  we  regard  the  written  signs  in  the  first  expres- 
sion as  signs  of  operation  or  as  signs  of  quality. 

Hence,  in  the  abridged  notation,  the  written  signs  in  any- 
polynomial  can  be  regarded  either  as  signs  of  operation  or 
as  signs  of  quality. 

When  all  the  written  signs  are  regarded  as  signs  of  qual- 
ity any  polynomial  becomes  a  sum. 

E.g.,  -_5  +  3-2=(-5)  +  (+3)  +  (-2). 

or  the  sum  of  the  terms  —  5,   +3,  and  —  2. 

Again,       7ac-4x  +  3y  =  -i-7ac+(-4)x  +  (+  3)?/, 
or  the  sum  of  the  terms  +  7  ac,   —  4  x,  and  +  3  y. 

In  general  formulas,  such  as  {A),  (B),  etc.,  it  is  usuall}- 
better  to  regard  the  written  signs  as  signs  of  operation ;  but 
in  most  other  cases  it  is  preferable  to  regard  the  written 
signs  as  signs  of  quality  and,  therefore,  to  regard  every 
polynomial  as  a  sum. 

58.  Coefficients.  If  a  term  is  resolved  into  two  factors, 
either  factor  is  called  the  coefficient,  or  the  co-factor,  of  the 
other. 

E.g.^  in  4  a&c,  +  4  is  the  coefficient  of  a&c,  +  4  a  of  &c,  +  4  a&  of  c, 
dbc  of  +  4,  and  6a  of  +  4  c. 

A  numeral  coefficient  is  a  coefficient  expressed  entirely  by 
numerals,  and  a  sign  of  quality  written  or  understood. 

A  literal  coefficient  is  a  coefficient  which  involves  one  or 
more  letters. 

E.g.^  in  —4xy,  —4  is  the  numeral  coefficient  of  xy ;  x  is  the 
literal  coefficient  of  —  4  ?/,  y  of  —  4tx,  and  —4x  of  y. 

When  in  a  term  no  numeral  factor  is  written,  1  is  understood,  e.g., 
a  denotes  -h  1  •  a  and  —  a  denotes  —  1  •  a  ;  abc  denotes  +  1  •  abc 
and  —  abc  denotes  —  1  •  abc. 


MULTIPLICATION  OF  HEAL  NUMBERS  45 

Exercise  18. 
Find  the  value  of  each  of  the  following  expressions : 

1.  15-9.  5.    (-Il)x7. 

2.  -9+7.  .  6.    (-7) -(-4). 
3.-8-6.                          7.    9-74-4-3  +  5. 

4.    (-3)  (-4)  8.    18-(-3)x(-4)-8. 

9.   35_j.(_7)x6-r-15x(-2). 

Find  the  value  of  a-[-h  —  c-\-d  and  a—{—h  +  c  —  d). 

10.  When  a  =  2,  6  =  —  4,  cz=^—Q>,  d=  —7. 

11.  When  a=  —7,  b  =  —  S,  c  =  5,  d  =  —  6. 

Find  the  value  of  x(y  —  v-\-  z). 

12.  When  x  =  6,  y  =  —  7,  v  =  — 9,  2  =  8. 

13.  When  a;  =  —  5,  y  =  l^j  v  =  —  4,  2  =  —  7. 

Find  the  value  of  x-T-{y  —  v  —  z). 

14.  When  x  =  -10,  y  =  -Q,  v  =  -9,  z  =  S. 

15.  When  a;  =  -16,  2/ =  -  10,  ^  =  -12,  z  =  6. 

16.  What  is  the  coefficient  of  a  in  a?  In  —a?  In 
-7ay? 

17.  In  the  expression  —Sab(x  —  y\  what  is  the  coeffi- 
cient of  a;  -  2/ ?     Oib(x-y)?    Of  8a?     Oi -S(x-y)? 

18.  If  the  sum  (x  —  y) -\- (x  —  y) -{-  (x  —  y)  -\ to  a  sum- 

raands  is  expressed  as  a  product,  what  is  the  coefficient  of 
x-y? 

59.  Having  given  a  product  and  one  factor,  division  is  the 
operation  of  finding  the  other  factor.  That  is,  if  n  is  one 
factor  of  m,  m  -^  n  denotes  the  other  factor ;  whence 

(m-i-n)  X  n=  m.  (1) 


46  ELEMENTS   OF  ALGEBRA 

60.  The  distributive  law.  The  product  of  a  polynomial  by 
a  monomial  is  equal  to  the  sum  of  the  products  obtained  by 
multiplying  ea,ch  term  of  the  polynomial  by  the  monomial; 
and  conversely. 

That  is,      {a-{-b-\-c-{-"')x  =  ax-{-bx  +  cx+"'        (0) 

The  distributive  law  lies  at  the  basis  of  multiplication  in  Arithmetic, 
e.g.^  if  we  wish  to  multiply  any  number  as  248  by  7,  we  separate  248 
into  the  parts  200,  40,  and  8,  multiply  each  of  these  parts  by  7  and 
add  the  results. 

Thus,  248  X  7  =  (200  +  40  +  8)  x  7  (1) 

=  200  X  7  +  40  X  7  +  8  X  7  .  (2) 

=  1400  +  280  +  56  =  1736. 
We  pass  from  (1)  to  (2)  by  the  distributive  law  (C). 

Prove  each  of  the  following  particular  cases  of  (O): 

Ex.  1.     (4-3  +  5). (-2)  =  4(-2)  +  (-3).(-2)4-5(-2). 

Ex.  2.  (-  4  +  2  -  6)(-  3)  =  (-  4).(-  3)+  2  (-  3)  +  (-  6).(-  3). 

Ex.  3.  («  +  6  +  c)-3  =  3a  +  36  +  3c. 

Proof     Let  the  multiplicand  be  any  binomial  a-\-b. 

The  proof  involves  three  cases :  when  the  multiplier  is 
(i)  a  positive  integer,  (ii)  a  positive  fractional  number, 
(iii)  a  negative  number. 

(i)  Let  m  be  any  positive  whole  number ;  then 
(a  -h  6)  m  =  (a  +  6)  -h  (a  +  6)  H —  to  m  summands  §  47 

=  (a  +  a  H —  to  m  summands) 

+  (6  +  6  H —  to  m  summands)     §  36 
=  am-\-bm.  (1) 

(ii)  Let  m  and  n  be  any  positive  whole  numbers  other 
than  zero;  then  —  will  denote  any  positive  fractional 
number. 


MULTIPLICATION  OF  REAL  NUMBERS  47 

(a  +  b)  (m  ^  n)7i  =  (a  +  6)m  §§  49,  59 

=  am  -\-  hm.  by  (1) 

=  a{m--rn)n-\-h(m^n)n  §§  49,  59 

=  [a  {m  -T-n)-\-b  (711  h-  n)']  n.  by  (1) 

Dividing  the  first  and  last  expressions  by  n,  by  (v)  of 
§  32  we  obtain 

(a  +  b)  (m  -J-  n)  =  a(m-i-  n)  -\-  b  (m  h-  n).  (2) 

Let  r  be  any  positive  number,  whole  or  fractional ;  then, 
from  (1)  and  (2)  we  have 

(a  -\-b)r  =  ar  +  br.  (3) 

(iii)   If  the  quality  of  equal  numbers  is  changed  from  -f 
to  — ,  or  from  —  to  +,  the  resulting  numbers  will  be  equal. 

Henee,  changing  the  quality  of  both  members  of  (3)  we 
have 

(a  +  6)  (-  r)  =  a (-  r)  +  6  (-  r),         §§  54,  55 

where  —  r  is  any  negative  number,  whole  or  fractional. 

A  similar  proof  would  apply  to  any  polynomial  as  well 
as  to  a  +  6 ;  hence  the  law  as  stated  in  (C). 

Ex.  1.   Multiply  3  a2  -  5  a  +  3  6  by  2  X. 
(3  a2  -  5  a  +  3  6)(2  a;)  =  (3  a2)(2  a;)  +  (-  5  a)(2  x)  +  (S  6)(2  x) 

=  6a2x-  10ax  +  6bx. 
Observe  that  in  applying  (  C)  we  regard  a  polynomial  as  a  sum. 

Ex.  2.   Multiply  2  x^  -  3  x2  -  2  x  by  -3  a. 

(2x3-3x2-2x)(-3a)=(2x8)C-3a)  +  (-3x2)(-3a)  +  (-2x)(-3a) 
=  -  6  ax3  4-  9  ax2  +  6  ax. 


48  ELEMENTS   OF  ALGEBRA 

Exercise  19. 
Multiply : 

1.  a;  +  2  by  3.  6.    2  cy  -Ax  hj  -a. 

2.  6a-75by-2.  7.    2  a- 3  & -c  by  -  2  a;. 

3.  2  a;  -  6  by  -  5.  8.    -  3  a;  +  2  ?/  -  5  2;  by  3  a. 

4.  2x-5by-3a.  9.    a;^- 3  a; +  4  by  -2  a. 

5.  ax-3b  hj  -2c.  10.    ar'-2  ?/-3  2;  by  -  5  a. 
11.    -2x'-hSxy-4:i/-x-{-2y-7  hj  -3a, 


CHAPTER   IV 

ADDITION  AND  SUBTRACTION  OF  INTEGRAL 
LITERAL  EXPRESSIONS 

61.  An  integral  literal  expression  is  an  expression  which 
involves  only  additions,  subtractions,  multiplications,  and 
positive  integral  powers  of  its  letters. 

Any  expression  which  contains  a  literal  divisor  is  called 
a  fractional  literal  expression. 

E.g.,  a'^  +  f  and  4  x^  —  |  6<  are  integral  literal  expressions  ;  while 

-  and are  fractional  literal  expressions. 

y         4-6 

A  letter  can,  in  general,  denote  any  integral  or  fractional 
number ;  hence,  any  literal  expression  can  have  any  integral 
or  fractional  value. 

E.g.,  wlien  x  =  I  and  y  =  |,  the  integral  literal  expression 

x  +  y  =  ^+^  =  f,  a  fractional  number. 

Also,  when  a*  =  2  and  y  =  3,  the  integral  expression  |  xy  =  ^. 
Again,   when  x  =  10  and  y  =  2,   the  fractional  expression  ?  =  5. 

y 

Tlie  pupil  must  clearly  distinguish  between  integral  and  fractional 
expressions  and  integral  and  fractional  numbers. 

62.  Like  or  similar  terms  are  terms  which  do  not  differ, 
or  which  differ  only  in  their  coefficients. 

E.g.,  4  ab  and  4  ab  are  like  terms ;  so  also  are  4  ab  and  —  10  ab. 
Again,  6  axy  and  —  4  bxy  are  similar  terms,  if  we  regard  6  a  and 

—  4  &,  respectively,  as  the  coefficients  of  xy  in  the  two  terms ;  but  if 
6  and  —  4  be  taken  as  the  coefficients,  these  terms  are  dissimilar. 

49 


60  ELEMENTS   OF  ALGEBRA 

63.  Sum  of  similar  terms.  The  converse  of  the  distribu- 
tive law  in  §  60  is 

ax-\-hx-\-cx-\-  '••  =  (a-\-'b  -{-  C  +  '^•)x.  (O) 

That  is,  the  sum  of  two  or  more  similar  terms  is  equal  to 
the  sum  of  their  coefficients  into  their  co7nmon  factor. 

1.  Find  the  sum  of  7  a,  —  5  a,  4  a. 

(+  7)a  +  (-  5)rt  +  (+4)a  =  (7  -5  +  4)a^6a. 

2.  Find  the  sum  of  3  ah"^,  -  5  ah"-,  -  8  al)^. 

(+  3)  a6--2  +  (-  5)  a62  +  (-  8)  a62=  (3  -  5  -  8)  ah'^=  -  10  db^. 

3.  Find  the  sum  of  7  (a  —  6) ,  —  5  (a  -  6),  4  (a  -  &). 

(  +  7)(a-6)  +  (-5)(a-5)  +  (+4)(a-&)=(7-5+4)(a-6)  =  6('?'-6). 

64.  By  §  57  the  sum  of  two  or  more  terms  is  indicated  by 
writing  them  in  succession,  each  term  being  preceded  by 
the  sign  of  quality  of  its  numeral  coefficient. 

The  sum  of  unlike  terms  can  only  be  indicated. 

E.g.,  the  sum  of  —  5  c,  7  a,  and  —  9  6  is 

-5c  +  7a-9&,  or  7rt-5c-9?). 
Again,  the  sum  of  —  3  ax,  —  5  by,  and  6  cz  is 

—  ^ax  —  ^hy-^Qcz,  or  6  C0  —  3  ax  —  5  by. 

Exercise  20. 
Find  the  sum  of : 

1.  2  a,  -3  a,  5  a.  6.  4  aft^,  -1  ab\  3ab\ 

2.  —4:X,2x,—x.  7.  —Sx%  5x%  —4  a;". 

3.  ab,  -2  ah,  3  ah.  8.  2  oc^,  -  5  ac^,  -  8  acl 

4.  2  a*,  -3a^  1  a\  9.  -5aV,  -3aV,  9  aV. 

5.  a;",  —2  a;'*,  4  a;".  10.  4&"2/'"^  —  Td^y™,  9  5"2/'"' 


ADDITION  AND  SUBTRACTION  51 

11.  7aa^,  —  5aic^,  4aa^,   —9aa^,  —  14aaj^,  25  aa?^. 

12  9aa^,  —  aar',  4aic^,  —Ihx^,  —14  car'. 

13.  -3^2/2!,  f  a^2;,  -^xyz,  6xyz,  -^xyz,  -^^-xyz. 

14.  (x-af,  -2(x-a)\  4.{x-af,  -5{x-a)\  12{x-aY. 

15.  (a^+2/2),  -5(a^+/),  9(a:^+yO,  -3(a^4-2/^,  -l{^+fr 

16.  (aT^-2/'),  -4(ar^-.v«),  _3(a:«-r),  -7(a^-2/«),  8(a^-2/3), 

Simplify  each  of  the  following  expressions  by  combining 
like  terms : 

17.  a^_7a^+4a^-5a^.  19.    o?y--^a?f+4.a?f-l^y\ 

18.  a;"— Sx^+Saj"— 7  a;".  20.    aar'— 7a.-2+6a:2_5^ 

21.  _9ic2_^i7^_^3a.o_^^_^^^2_5^2 

22.  Sab' -7 aV  +  8 a/>-  - 4 ah'' -\- 7  ca^ -  11  ca^. 

23.  -12a^-^4:a^-9x^-\-7a^  +  Sa^-9a''  +  7a\ 

24.  7  aftcc?  —  11  abed  +  41  a6cd  +  7xy  —  20xy. 

25.  _5aj2_2a:2  +  |a^  +  8/--|/-f/. 

26.  7ar'  +  2a2-5.'c2-3a2. 

7  a:2  +  2  a2  -  5x2  -  3  a2  =  7  a;2  _  5x2  +  2  a2  -  3  a2      by  (^) 
=  2x2-a2. 

27.  7a6  — 5a;2/  +  3a6 +  2a^  — 6a5  — a^. 

28.  -9ax'-\-5bf-\-7ax^-3bf+llaa^  +  4:bf. 

29.  —  7  c/  —  4  a6  +  9  a.-2;  4- 11 C2/2  + 10  a6  —  5  a;2;  —  a6. 

30.  2(a^-l)  +  3(a2  +  l)-4(x2-l)-5(a2  +  l). 

31.  3(a'  +  b')-4.(x-\-y)-7(a'-\-b')-^5(x-\-y), 

32.  Review  this  exercise,  solving  each  example  mentally. 


52  ELEMENTS   OF  ALGEBRA 

65.  Addition  of  polynomials. 

Ex.  1.     Add  -  3  x^  +  7  X  to  5  a;2  _  4  a;. 
(5  a:2  -  4  x)  +  (-  3  x2  +  7  x)=  5  x2  -  4  x  -  3  x-  +  7  x  by  converse  of  {B) 
=  2x2  +  3x.  by  (^),  (S) 

Ex.  2.     Find  the  sum  of 

4  x2  -  3  xy  +  y2,  _  2  x2  -  5  xy  -  6  2/2,  and  2  x?/  -  x2  -  3  62. 

In  adding  polynomials,  it  is  convenient  to  write  them  under  each 
other,  placing  like  terms  in  the  same  column. 

Thus,  (4  x2  -  3  xy  +  2/2)  +  (  _  2  x2  -  5  x?/  -  6  2/2)  +  (2  ccy  -  x2  -  3  h'^) 

can  be  written  4  x2  —  3  xy  +     2/^ 

-  2  x2  -  5  xi/  -  6  2/2 

-  x2  +  2x2/  -3  62 
x2  -  6  X2/  -  5  2/2  -  3  62. 

Here  the  rows  of  terras  are  the  groups  of  terms  as  given,  while  the 
columns  of  terms  are  the  groups  of  similar  terms  obtained  by  rearrang- 
ing and  regrouping  by  laws  {A)  and  {B). 

Since  there  is  no  carrying  as  in  Arithmetic,  the  addition  can  be 
performed  from  left  to  right,  or  from  right  to  left. 

66.  When  in  a  polynomial  the  exponents  of  some  one 
letter  increase  or  decrease,  from  term  to  term,  the  polyno- 
mial is  said  to  be  arranged  in  ascending^  or  in  descending^ 
powers  of  that  letter. 

This  letter  is  called  the  letter  of  arrangement. 

E.g.,  the  polynomial  x^  +  2  x^y  +  3  xy'^  +  4  2/^  is  arranged 
in  descending  powers  of  x,  x  being  the  letter  of  arrangement ; 
or,  in  ascending  powers  of  2/,  y  being  the  letter  of  arrangement. 

In  arranging  a  polynomial  in  ascending  or  descending 
powers  of  any  letter,  we  must  first  combine  all  the  terms 
which  contain  the  same  power  of  that  letter. 

In  adding  polynomials,  it  is  usually  convenient  to  arrange 
them  in  ascending,  or  descending,  powers  of  some  letter, 
as  below: 


ADDITION  AND  SUBTRACTION  63 

Ex.  1.    Find  the  sum  of  2  x^  -  3  x^  +  y,  _  4, 

7  X  -  4  x2  4-  5  x3  +  5,  and  7  x^  -  4  x^  +  2  x  -  1. 
Arranging  each  polynomial  in  descending  powers  of  x,  we  have 

-3x3  +  2x2+      x-4 
6x3-4x2+    7x  +  5 
-4x3  +  7x2+    2x-l 
-  2  x3  +  5  x2  +  10  X 

Exercise  21. 
Find  the  sum  of  : 

1.  a  +  26-3c,  -3a  +  6  +  2c,  2a-3&  +  c. 

2.  -Zx-\-2y-\-z,  x-^y-\-2z,  2x  +  y-Sz. 

3.  -15a-196-18c,   14a4-1564-8c,  o  +  56  +  9c. 

4.  5  aa;  —  7  6?/ +  C2;,  aa;  +  2  61/ —  02;,   —  3ax  + 2&2/  + 3c2;. 

5.  20 p  +  q-r,  p-20q  +  r,  2^ -\-q- 20 r. 

6.  —  5  a&  +  6  6c  —  7  ac,  8  a6  —  4  6c  +  3  ac,  —  2  06  —  2  6c 
4-  4  ac. 

7.  pq  +  qr  —  pTf  —  pq -\- qr -{- pr^  pq  —  qr -{- pr. 

8.  2  a6  4-  3  ac  4-  6  a6c,   —  5  a6  +  2  6c  —  5  a6c,  3  a6  —  2  6c 
—  3ac. 

9.  x^  +  xy  —  y-,  —z^-\-yz-\-  /,  xz  -\- z^  —  0^. 

10.  5a»-3c»  +  d3,  fo3_2a3  4-3^3,  4c3 -2a'»- 3d». 

11.  a,^  +  2/2_2ic?/,  2z2_3/-42/«,  2  ar^  -  2  2:2  -  3  ii'2;. 

12.  a.'3  +  3ar^2/  +  3a^',  -  3  ar^3/ -  6  a^/ -  a^,  3  a;^^,  +  4  a^. 

13.  x'-^x'^y-hx'f,  3a;^+2a.V-6an/*,  Zs^f+Qxy'-f. 

14.  a«-4a26  +  6a6c,  a-6  -  10  a6c  +  c^,  63  +  3a26  +  a6c. 

15.  3a2-1062  +  5c2-76c,        -  a2  +  462- IOC24- 3a6, 
c2+116c+8ac-2a6,  4c2-46c  +  ac,  -2a2+662-9ac-6c. 


54  ELEMENTS   OF  ALGEBRA 

16.  4ic*  +  12ar^-a;-10,  llx" -2x^  -  x' -\-'d,  ^x" -3x' 
+  4.X,  4ar^-x^-5,  Q,x^  -  si? -\-2x- -1. 

17.  i:^-\x  +  \,  -^x'  +  lx-^,  Ix^  +  lx^^. 

18.  la'  +  ^ah-^h^  lo?-ah-\b'',  -  a^- f  a6 +  2  6=^. 

19.  _2^_a;2/  +  2/^3x2-|a?y-i2/',  -  f  a^  +  2x'?/- |/. 

20.  -f  a^-|a;/  +  22/^,  |aj22^-|_aj?/2_|_i.2^a  |ar''-2a;2^-f ^Z'. 

21.  a^-3ax2_|_5^'j^_^3^2a^  +  4ax2-6a2a;,  Gaa^-Sa^a? 
+  a^,  —  2ar^  +  4a^a;  — 5al 

22.  3a;2  4-/-3?/2-2^  2  a;^/ -  3  2/^  +  3  2/^,  -4.x^-2xy-{- 

23.  Given  a;  =  6  +  2c  — 3a,  y  =  c  +  2a  —  Sh,  and 

2;  =  (x-f26  —  3c;  show  that  x-\-y  -\-z  =  (). 

24.  Given  a  =  5a;  —  32/  —  2  2;,  b  =  5y  —  3z  —  2x,  and 

c  =  5z  —  Sx--2yj  show  that  a  +  6  +  c  =  0. 

67.  To  subtract  one  expression  from  another,  change  the 
sign  before  each  term  of  the  subtrahend  from  +  to  ^  or  from 
—  to  +,  and  add  the  result  to  the  minuend. 

Proof.  Changing  the  sign  before  each  term  of  the  sub- 
trahend changes  the  quality  of  the  subtrahend  (§  55) ;  and 
by  §  40  the  minuend  plus  the  subtrahend  with  its  quality 
changed  is  equal  to  the  remainder. 

Ex.  1.   From  —  5  x'^y  take  4  x'^y. 

-  5  ic2?/ -  (  +  4  a;2i/)  =  -  5  ic2y  +  (  _  4  a^Zj/) 
=  -  9  x'^y. 

Ex.  2.   From  5  x-  +  a;?/  -  m  take  2  x^  +  8  xy  -  7  y2. 

Changing  the  sign  before  each  term  of  the  subtrahend  from  +  to  - 
or  from  —  to  +,  and  adding  the  result  to  the  minuend,  we  have 

5  a;2  +    xy  —  m 
-  2  x2  _  8  xy  +  7  1/2 

3  x2  —  7  xy  —  m  +  7  ?/2^  Remainder. 


ADDITION  AND   SUBTRACTION  55 

Note.  The  signs  of  the  subtrahend  need  not  be  actually  changed  ; 
the  operation  of  changing  the  signs  ought  usually  to  be  performed 
mentally,  as  in  the  following  example. 

Ex.  3.  From  2x*  -Sx^  +  7  x-S  take  x*  -2x^-9x  +  ^, 

2x*^  -Sx^-\-    7x-   8 

x^-2x^  -9x+4 


x*  +  2  a:8  -  3  a;2  +  16  a;  -  12 


Exercise  22. 

1.  From  4 a  —  3 6  +  c  subtract  2a  —  Sb  —  c. 

2.  From  15x -\-10y —  18z  subtract  2x  —  Sy  +  z, 

3.  From  — 10  be -\- ab  —  A  cd  take  —  11  ab-\- 6  cd. 

4.  From  ab  -\-  cd  —  ac  —  bd  take  ab  -\-  cd-^  ac  +  bd. 

5.  From  m^  +  Sn^  subtract  —4:m^—6n^-{-71x. 

6.  7xy-(-Sxy)  =  ? 

7.  -9x'y-(-\-6x'y)-(-20x'y)  =  ? 

8.  32a^-(-122/^-(-hl4a^)-(+92r)  +  (-2/)  =  ? 

9.  28  a'b'  -  (+ 17  a'6-)  -  (-  19  ar^y)  -  (+  15  a^y) 

-(-5a'b')=:? 

From 

10.  -8a^f-\-15x^y  +  lSxf  take  4:a^i/^-{-Txh/-Sxf. 

11.  a^bc  +  b^ca  +  c^a6  take  3  a^bc  —  5  &-ca  —  4  c^a6. 

12.  -7a264-8aZ>2  +  cd  take  5arb -7  ab' +  6cd. 

13.  10a262-|.i5a?>2_^3^25  take  -  10  a-^^  + 15  oft^  -  8  a^fe 

14.  hs^c^-2  abc  take  a^-{-b^-S  abc. 

15.  7abc-Sa^-\-5b^-(^  take  a^  +  6^  +  <?' - 3 «6c. 

16.  ix'-^xy-^  2/2  take   _  f  a;-  +  ;^  -  y-. 


56  ELEMENTS  OF  ALGEBRA 

17.  f  3^  — faa;  take  \  —  \x^  —  ^ax. 

18.  ^o? —  2aQi? —  \a^x  take  \  o?x -\- \  a^  —  ^  ax^. 

If  ^  =  a2_4^5_352^  J5=a6-462-3a^ 

find  the  expression  for 

19.  A  +  B+C+D.  22.  J._5-C-Z). 

20.  A  +  B-\-C-D.  23.  -^-J3H-C+^. 

21.  A^B-C-D.  24.  _^  +  5-6'  +  Z>. 

In  solving  example  20,  under  the  values  of  J.,  B,  and  (7  write  that 
of  D  with  its  quality  changed,  and  then  add  the  results. 

25.  From  5a^  +  3a;  — 1  take  the  sum  of  2x  —  5-{-7 a^ 
and  3aj2-f4-2a^  +  a;. 

26.  From  the  sum  of  2  a^  -  3  a^  +  a  -  2  and  2  +  8  a^  -  a^ 
subtract  3  a  —  7  a^  +  5  al 

27.  From  the  sum  of  4:a^  +  3x-7,  2af  ~Sx  +  2x^~l, 
and  — 5a^H-2a7  —  ar^  +  9  take  the  sum  of  2 aj^  —  11  ic  and 
9a^  +  5a^  +  3-2a;. 

68.  Removal  of  signs  of  grouping.  The  converse  of  the 
associative  law  for  addition  in  §  36  is 

a-^(b-\-c)  =  a  +  b-\-c.  (1) 

That  is,  a  sign  of  grouping  preceded  by  the  sign  +  can  be 
removed  if  each  enclosed  term  is  left  unchanged. 

Observe  that  the  sign  +  is  understood  before  b  within  the 
parentheses. 

Ex.  1.    a  +  (4  a  -  7  1/  +  5  «)=  a  +  4x-7y4-5«. 
Ex.2.    z+(^-Sx-{:2y  -ia)=z-Sx-\-2y-ia. 


ADDITION  AND  SUBTRACTION  57 

By  the  rule  for  subtraction  in  §  67,  we  have 

A  sign  of  grouping  preceded  by  the  sign  —  can  be  removed, 
if  the  sign  before  each  enclosed  term  is  changed  from  +  <o  — , 
or  from  —  to  ■}-. 

Ex.1.   5a-(3&-2a  +  4c)=5rt-36  +  2a-4c  (1) 

=  7a-35-4c. 
The  sign  +  is  understood  before  3  b  within  the  parentheses. 
Ex.2.    -(5?rt-4n)-(-3wi+7n)=-5m+4n  +  3m-7n        (2) 

=  -  2  m  -  3  71. 

69.  Sometimes  one  sign  of  grouping  is  enclosed  within 
another ;  in  this  case  the  different  signs  of  grouping  must 
be  of  different  shapes  to  avoid  confusion. 

When  there  are  several  signs  of  grouping  they  can  be 
removed  one  at  a  time  by  the  rules  of  §  68 ;  and  it  is  better 
for  beginners  to  remove  at  every  stage  the  innermost  sign  of 
grouping. 

Ex.    Removing  the  signs  of  grouping,  simplify  the  expression 

a-lx  +  {y-ib-c)]-zl 

a-  [x  +  {y-(&-c)}-«]  =  a-  [x  +  {y-6  +  c}-2] 

=  «  -  [x-\-y  —  b-{-c-z'\ 

=  a —  x  —  y  +  b  ~  c  +  z.  (1) 

In  the  above  process  the  parentheses  (  )  were  removed  first,  Uien 
the  braces  {  },  and  then  the  brackets  [  ]. 

Verify  (1)  when  a  =  8,  x  =  3,  y=-2,  &  =  -3,  c  =  -4,  z  =  7. 

Removing  the  outer  sign  of  grouping  first,  we  have 

«-  [«  +  {y-C&  -  c)}  -  z^=a  -  X  -  {y  -{b  -  c)}  +  z 
=  a  -  X  —  y  -^  (b  -  c)  -{-  z 
=  a-x  —  y-\-b-c-^z. 

In  review,  the  student  should  begin  with  the  outer  sign  of  grouping, 
as  he  can  thereby  soon  learn  to  remove,  without  error,  two  or  more 
signs  of  grouping  at  a  time. 


58  ELEMENTS  OF  ALGEBRA 

Exercise  23. 

Simplify  each  of  the  following  expressions  by  removing 
the  signs  of  grouping  and  combining  like  terms : 

1.  a-(6  +  c)  +  (&-c-a). 

2.  Sx-(y-2x)  +  (z-{-y-5x), 

3.  z-\?j-(z-x)\. 

4.  3x-\2y  +  5z-{Sx^y)l. 
5  a—[a  —  \a—(2a— a)]']. 

Verify  the  results  of  examples  1  to  5  inclusive, 

6.  When  a  =  7,  b  =  -3,  c  =  4.,  x=10,y  =-5,  z  =  -2. 

7.  When  a  =  —  5j  b  =  2,  c  =  — 1,  x=—3,  y=^,  z=—7. 

8.  a  +  b  —  [a  —  b-\-la-\-b  —  {a  —  b)\']. 


9.    x-(y-z)-\-l2z-3y-5xl. 


10.  2a-{36  +  (4c-36  +  2a)|. 

11.  a-2b-\3a-{b-c)-5c\. 

12.  a-l3b-^\3c-(d-b)  +  a}-2a']. 


13.  2x-(5y-3z-]-7)-l4.-{-\x-(3y  +  2z-\-5)l]. 

14.  3a  -  [26- J4c- 12a- (4  ft- 8c) J -(6 6 -12c)]. 

15.  -  115 X -{Uy -  (15z  -^12y)  -  (10 X-  15  z)}']. 

16.  —  [a  — 5a  +  (ic— a)  —  (ic— a)  — aj  — 2  a]. 

17.  2x-(3y-4.z)-\2x-(3y-\-4.z)\-  \3y-{4:Z+2x)\. 

70.   Insertion  of  signs  of  grouping.     Law  (B)  in  §  36  is 

a  +  6  4-  c  =  a  +  (6  +  c). 

That  is,  a7iy  number  of  terms  of  a  polynomial  can  be  enclosed 
within  a  sign  of  grouping  preceded  by  the  sign  -[-,  if  each 
enclosed  term  is  left  unchanged. 


ADDITION  AND  SUBTRACTION  69 

Ex.  1.     5x-7?/  +  4c-7&  =  5a;  +  (-7y  +  4c-7  6). 
Ex.  2.     4a  +  3c-5x-3y  =  4a  +  (+3c-5x-3y). 

From  the  rule  for  subtraction  it  follows  that, 

Any  number  of  terms  of  a  polynomial  can  be  enclosed  within 
a  sign  of  grouping  preceded  by  the  sign  —,  if  the  sign  before 
each  enclosed  term  is  changed  from  -\-  to  —  or  from  —  to  +. 

Ex.  1.  7x  +  6y-5a  +  7c  =  7a;-(-6?/  +  5a-7c). 

Ex.  2.  ax^  —  2  ex  —  cx^  -\-  hx^  —  x  =  ax^  —  cx^  +  fex^  —  2  ex  —  x 

=  (a  -  c)x3  +  6x2  _  (2  ex  +  x) 
=  (a  -  c)x3  +  6x2  -  (2  c  f  i)a;. 

Exercise  24. 

In  each  of  the  following  expressions  enclose  the  last  four 
terms  within  a  sign  of  grouping  preceded  by  the  sign  — , 
without  changing  the  value  of  the  expression : 

1.  3x  — 2a  +  56  — 2/  +  2;. 

2.  a-b-x  +  Zb-z-\-2y. 

3.  Sy-\-2x  +  lz  +  a-\-2b-\-c. 

4.  2z-7a;-2a-36-5c -9?/. 

Simplify  each  of  the  following  expressions  by  combining 
the  terms  having  the  same  powers  of  x,  so  as  to  have  the 
sign  +  before  each  sign  of  grouping : 

5.  ax^  +  by?  +  ^-\-2bx-b^-\-2x^-Zx. 

Ans.  {a^2)x'  +  (b-b)x'  +  (2b-^)x  +  5. 

6.  3  6ar^  —  7  —  2  ic  +  a5  +  5  aa;'  +  co;  —  4  0^  —  6af. 

7.  2-7ar'  +  5ax2_2ca;  +  9ax'+7aj-3a:2 

8.  2  cy^  —  ^  abx  -\-  ^dx  —  3  bx"^  —  aV  +  x\ 


60  ELEMENTS   OF  ALGEBRA 

Simplify  each  of  the  following  expressions  by  combining 
like  terms  in  x  so  as  to  have  the  sign  —  before  each  sign  of 
grouping : 

9.  aa^-\-5a^-a^x*-2b:x^-Sx'-bx\ 

10.  7  a^  —  3  c^x  —  dhx'  +  5  aa;  +  7  a^  —  ahca^, 

11.  ay? -{-o?^ —  h^ —  hy?  —  cx^. 

12.  ?>}y^x^ —  hx  —  ax^ —  cx^ —  h(?x  —  l  x^. 

Simplify  the  following  expressions,  and  in  each  result 
add  the  terms  involving  like  powers  of  x\ 

13.  a:(?—2cx—{bQi?—  \GX—dx—il)x^-\-Z c^)\  —  {c^—hx)\. 

14.  5 aa^  -  (7  6aj  -  7  ca;-)  -  J6  6a^  -  (3  aa;^  +  2  aa;)  -  4  cx'X. 

Express  in  descending  powers  of  x  the  sum  of, 

15.  a^ci?—5x,  2ax^—5aa^,  2x^  —  bx'^  —  ax. 

16.  aa?^bx  —  c,  qx  —  r—py?,  ar4-2a;  +  3. 

17.  pa?  —  qx,  qa?  —px,  q  —  x^,  jyy?  +  qm?. 

18.  2aa^-3ca?2+^a;,  3y?a;- ma^-2ca^,  a;-2a^-3a?'. 

19.  bx  —  ay?  —  bo?,  Sx^ —  4:nx  —  2ma?,  2a?— po?. 

20.  coi?^2ax-\'mo?,  4:X^'-bx^,  4:7ix-\-2px\   3a?—2rx^~x. 


CHAPTER  V 

MULTIPLICATION  OF  INTEGRAL  LITERAL 
EXPRESSIONS 

71.  The  degree  of  an  integral  term  is  the  number  of  its 
literal  factors.  But  we  usually  speak  of  the  degree  of  a 
term  in  regard  to  one  or  more  of  its  letters. 

E.g.^  5  ax  is  of  the  second  degree,  and  7  a^^a  is  of  the  fifth  degree. 
Again  4  abx-y^,  which  is  of  the  seventh  degree,  is  of  the  first  degree  in 
a,  of  the  second  degree  in  x,  of  the  third  degree  in  y,  and  of  the  fifth 
degree  in  x  and  y. 

72.  The  degree  of  a  polynomial  is  the  degree  of  its  term 
of  highest  degree. 

E.g.,  the  trinomial  ax"^  +  bx -\-  c  is  of  the  first  degree  in  a,  6,  or  c, 
and  of  the  second  degree  in  x.  The  binomial  ax^y  +  by-,  which  is  of 
the  fourth  degree,  is  of  the  second  degree  in  x  or  y,  and  of  the  third 
degree  in  x  and  y.  The  trinomial  ax^  +  2  bxy  -f  cp^  is  of  the  second 
degree  in  x,  in  y,  and  i?i  x  and  y. 

73.  An  expression  is  said  to  be  homogeneous  in  one  or 
more  letters  when  all  its  terms  are  of  the  same  degree  in 
these  letters. 

E.g.,    2a^  -\-Sab  +  4b^  is  homogeneous  in  a  and  b  ; 

5  x^  +  3  x:^y  -{-Sxy^  +  y^  is  homogeneous  in  x  and  y  ; 
and  0x2  ^  2  bxy  +  cy^  is  homogeneous  in  x  and  y. 

Exercise  25. 
What  is  the  degree  of  the  term  3  arba^y*, 

1.  In  a?     3.    In  a;?      5.    In  a  and  6?     7.    In  a,  a?,  and  ?/ ? 

2.  In  6?     4.    In  y?     6.    In  x  and  y?      8.    In  b,  x,  and  y? 

61 


62  ELEMENTS   OF  ALGEBRA 

What  is  the  degree  of  the  trinomial 

a'x'-\-7a'b'i^y'-5abxy% 
9.    In  a;?     10.    In  a?     11.    In  a;  and?/?     12.   In  6  and  y? 

Write  two  trinomials  of  the  third  degree  and  homogeneous, 
13.    In  a  and  b.  14.    In  x  and  y.  15.    In  a  and  x. 

74.  A  product  is  zero  when  one  of  its  factors  is  zero. 
That  is,  a  .  0  =  0  and  0  .  a  =  0. 

Proof  a'0  =  a(b-h),  §§  11,  32 

=  ab-ab  =  0.  §§60,11 

Similarly,  0  •  a  =  (6  —  6)  a  =  0. 

Conversely,  when  a  product  is  0,  one  or  more  of  its  factors 
isO. 

That  is,  if  a  •  6  =  0,  then  a  =  0,  or  6  =  0,  or  a  =  0  and 
6  =  0. 

75.  Any  positive  integral  power  of  0  is  0  ;  that  is  0"  =  0. 
Proof  0"  =  0  •  0  .  0  .. .  to  ?i  factors  =  0.  §  74 

76.  Product  of  powers  of  same  base. 

Ex.  1.   23  X  22  =  (2  X  2  X  2)(2  x2)  =  2  x  2  x  2  x  2  x  2  =  2^. 

Ex.  2.   a^a^  =  (aaa)  (aa)  =  aaaaa  =  a^. 

These  examples  illustrate  the  following  law  of  exponents. 

The  product  of  the  mth  poiver  and  the  nth  power  of  the 
same  base  is  equal  to  the  (m  +  n)th  power  of  that  base,  and 
conversely. 

That  is,  a'"  '  a"  =  a'"+". 

Proof  a'^a''  =  (aaa  •  •  •  to  m  factors)  (axxa  •  •  •  to  w  factors)  §  52 
=  aaa  •••  torn  +  91  factors  §49 

=  a"'+\  §  52 


MUL  TIP  Lie  A  TION  63 

Ex.   Multiply  3  a-x^  by  —  4  a^x^y. 

(3  aV)  (  -  4  a%2y)  =  s  a'^^^  {- i)  a^x^y  §  49 

=  3(-4).a2«4.a;3a;2.y  §49 

=  -  12  a^xSy.  §§  47,  76 

This  example  illustrates  the  method  of  finding  the 

77.  Product  of  two  or  more  monomials.  Using  the  commu- 
tative and  associative  laws,  we  have  the  following  rule : 

Multiply  together  their  numeral  factors,  observing  the  law 
of  quality  ;  after  this  write  the  product  of  their  literal  factors, 
observing  the  law  of  exponents, 

Ex.   Multiply  together  —  5  ay^,  —  2  a^x^,  and  —  9  az^y. 
(_  5 ay2)(_  2  a2x8)(_  9  axh,)  ==  (_  5)(-  2) (-  9)  aa^Q;x?xhpy 

=  -  90  a*x5y3. 

Exercise  26. 
Find  the  product  of : 

1.  a^  and  a*.  7.  —Za%  and  12a5«. 

2.  a^  and  a*.  8.  —  abed  and  —  3  aC-b'^c. 

3.  rf,  jf,  and  y^  9.  7  x^y'^7^  and  —  5  3?yh. 

4.  aa;  and  3  ax.  10.  —  3  a^6V  and  8  a^b^c*d. 

5.  —  2a6a;  and  —Tab.  11.  2  a^,  —  4  a^ft,  and  5ab^ 

6.  6  ic^y  and  —  10  axy.  12.  —  5  ax,  —  7  a%  and  2  aic^. 

13.  8  xy^,  —  3  ar^i/,  and  —  3  xy. 

14.  -  7  aft^,  _  3  a'b%  and  -  a^ftl 

15.  a^b%  2  aft'^c,  and  —  5  a6c. 

16.  —  7  x^y^,  ay^y*,  and  aa^. 

17.  —  a^bx,  aWx,  and  —  aa^. 

18.  —  a^ar,  —  b'^x,  and  —  a5?/. 


64  ELEMENTS   OF  ALGEBRA 

78.  Multiplication  of  a  polynomial  by  a  monomial.  The  dis- 
tributive law  of  multiplication  is 

(a  -f-  6  -f  c  -|-  •")x  =  ax  -\- bx  -\-  ex -\-  •••. 

That  is,  to  multiply  a  polynomial  by  a  monomial,  multi- 
ply each  term  of  the  polynomial  by  the  monomial,  and  add 
the  several  products. 

Ex.    Multiply  2  ^2  _  4  2,3  ijy  _  3  y^, 

=  _  6  ?/^3  _!_  12  yiz. 

Writing  the  multiplier  under  the  multiplicand,  ^ 

the  work  can  be  arranged  as  at  the  right  of  the       ~     ^^ 

page.  —Qyz^-\- 12  y^z 

Exercise  27. 
Multiply : 

1.  4a2-5a  +  3&  by  2  a\ 

2.  2a2  +  3a6  +  262  by  -Sd'b\ 

3.  bc-{-ca  —  ab  by  abc. 

4.  2a^-3x^-\-5x-4:hj  -Bx". 

5.  -4:X^-\-3a^-Sx^  +  4L  hj  -6i^. 

6.  9gh-12ga-3gb  by  Sgh. 

7.  —  a^6c  +  b^ca  —  c^ab  by  —  ab. 

8.  —  5  £C2/^2;  +  3  xyz^  —  8  x?y%—  7  a;^/^  by  —  2  a;^^;. 

9.  ci^ftV  -~  abc  —  ax —by  —  cz  by  —  5  abcxy. 

10.  f  aV  — f  aa^  +  ^aa;  by  —  |a^a;. 

11.  -|i»?/2^iaaJ2/-2a/  +  |a2^  by  -faa^. 

12.  i^a^y-fa^/H-iaa^-^a^/by  -i^a;^2/. 

13.  (x  +  yy-2a(x-\-y)-\-5a^  hj  2(x  +  y). 

14.  (aj  +  l)«-4a(a;  +  l)'-2a6  by  -5a6(aj  +  l)'. 

15.  (a'-hiy-Sx{a'  +  iy-4:xy  by  _  3  aj^^/ (a^ -f  1)4. 

16.  (a^  +  yy-  a(a^  -^yf  +  3  a'b^  by  - 4  a=^6* (a^  +  2//. 


MULTIPLICATION  66. 

Remove  the  signs  of  grouping,  and  simplify  each  of  the 
following  expressions  : 

17.  (a  +  h)c-(a-h)c.  19.    ^(6  -  2  c)  +  |(c- 2  6). 

18.  2(a-6)+4(a  +  6).  20.    1  aQ)  -  c) -2h{a- c). 

21.  a^h^c^-dF)  +  (^d?{a?-W)  +  hh\d'-aF). 

22.  2\Zab-4.a{c-2b)\. 

23.  7ac-252c(a-36)-3(5c-26)a[. 

79.   To  multiply  one  polynomial  by  another, 

Multiply  each  term  of  the  multiplicand  by  each  term  of  the 
multiplier  J  and  add  the  resulting  products. 

Proof  Let  x-\-y-^z  be  the  multiplicand,  and  a  +  &  the 
multiplier ;  then  by  successive  applications  of  the  distribu- 
tive law,  we  have 

{x-\-y  +  z)(a  +  b)  =  x(a-\-b)  +  y{a-^b)+z{a-^b) 

=  xa -\- ya -{- za -{- xb  +  yb  -\-  zb.       §  36 

Similarly  when  each  factor  has  any  number  of  terms. 

Ex.  1.     Multiply  -2x  +  3y  by  4x-7y. 

=(-2a;).4a;+3y.4a;+(-2a;)(-7y)4-3y(-7y)  (1) 

=  -  8  x2  +  12  xy  +  14  xy  -  21 2/2  (2) 

=  -  8  x2  +  26  xy  -  21  y2.  (3) 

Performing  tlie  steps  in  (1)  and  (2)  mentally,  we  can  arrange  the 
work  as  below : 

-2a;  +  3y 
4x  —  7y 

-  8  a;2  +  26  ary  -  21 2/2 

Observe  that  the  first  and  last  terms  in  the  product  are  the  products 
of  terms  in  the  vertical  lines,  while  the  second  term  is  the  sum  of 
the  products  of  the  terms  in  the  diagonal  lines. 

In  this  way  solve  the  first  15  examples  in  Exercise.  28. 


66  ELEMENTS  OF  ALGEBRA 

Ex.  2.     Multiply  a;^  -  2  jc  +  x^  +  1  by  2  -  x -\- x^. 

Arranging  both  multiplicand  and  multiplier  in  descending  powers 
of  ic,  we  have 

x^  +  x^-2x  +1  (1) 

x^-x  +2 

afi  +  x^-2x^+    a;'2  (2) 

_  a4  _     a;3  +  2  a;2  _     x  (3) 

2  x3  +  2  a;-2  -  4  X  +  2  (4) 

x5  -x^  +  5x'^-bx  +  2 

Expression  (2)  is  the  product  of  (1)  multiplied  by  x"^ ; 
Expression  (3)  is  the  product  of  (1)  multiplied  by  —  x  ;  and 
Expression  (4)  is  the  product  of  (1)  multiplied  by  2. 
The  sum  of  these  partial  products  is  the  required  product,  by  §  79. 

A  vertical  line,  or  bar,  is  often  a  convenient  sign  of  group 
ing.     Its  use  is  illustrated  in  the  next  example. 

Ex.  3.     Multiply  x^-2x^y  +  ^  xy'^  -  y^  by  x2  -  3  xy  +  ?/2. 


x3  -  2  x2y  +    3 
x^-Sxy  + 

XJ/2    -  y3 

y^ 

x5-2x*.v+    3 

-3        +6 

+    1 

x^y^-   1 

-  9 

-  2 

x2|/3 

+  3 

+  3 

xy* 
-y' 

x5  -  5  x^y  +  10 

x^y^  - 12 

X22/3+6 

xy^-y^ 

The  sum  of  the  numbers  before  each  bar  is  the  coefficient  of  the 
literal  factor  after  it. 

In  this  example  the  multiplicand  and  the  multiplier  are  both  homo- 
geneous.    Observe  that  the  product  is  homogeneous  also. 

This  illustrates  the  following  principle. 

80.  The  product  of  two  or  more  homogeneous  expressions  is 
homogeneous. 

Proof  If  the  homogeneous  multiplicand  is  of  the  nth 
degree,  and  the  homogeneous  multiplier  is  of  the  mth 
degree,  then  each  term  in  the  product  will  be  of  the 
(m  +  n)th   degree ;    that  is,  the  product  will  be  a  homo- 


M  UL  TIP  Lie  A  riON  67 

geneous  expression  of  the  (?>i  +  n)th  degree ;  and  so  on  for 
any  number  of  factors. 

When  the  multiplicand  and  the  multiplier  are  homogeneous,  that 
fact  should  be  noted  in  every  case  by  the  pupil ;  and  if  the  product 
obtained  is  not  homogeneous,  it  is  at  once  known  that  there  is  an  error. 

Exercise  28. 
Multiply : 

1.  x-{-2y  by  x  —  2y.  10.  —a; +  7  by  a;  — 7. 

2.  2x-\-Syhy3x  —  2y.  11.  —x—16  by  —x-\-16. 

3.  a  — 3 &  by  a +  36.  12.  —  a; +  21  by  a;  — 21. 

4.  x  +  7bya;-G.  13.  2a  +  |&  by  3  a  + J  &. 

5.  3a;-7  by  2a;-l.  14.  ^a- ^b  hj  ^a-^b. 

6.  2.0;  — 4  by  2a; +  6.  15.  ax  — by  by  ax-\-by. 

7.  22/  +  5^>  by  3?/  — 46.  16.  a.'2  +  a;  + 1  by  a;  — 1. 

8.  2  7/i2  +  5n2  by  2  m-  — ir.  17.  cr  +  a6  +  6-  by  a  —  b. 

9.  3m2-l  by  3?n-  +  l.  18.  a'--a6  +  6-  by  a  +  6. 

19.  x^-xY-\-y^  by  a^  +  2/^. 

20.  a^-ab-^b''  hy  o^-{-db-\-b\ 

21.  a^  _  2  ax  +  4  aj2  by  a^  +  2  aa;  +  4  x^. 

22.  10  a-  +  12  a6  +  9  6^  by  4  a  -  3  b. 

23.  a-x  —  a^  -\- si?  —  a?  by  x-\-a. 

24.  x^^x-2  by  x--\-x-Q,. 

25.  2a:3_3aj2^2a;  by  2a^  +  3a;  +  2. 

26.  a^  +  2a?b+2ab^  by  a^  -  2  ab -\- 2  b\ 

27.  Qi?  —  3xy  —  'ifhy—^-\-xy-\-y'^. 

28.  ar  -  2  a;?/ +  2/-  by  ar'  + 2  a.-?/ +/. 

29.  27a.-3-36aa:2^48a2a;-64a3  by  3a;  +  4a. 

30.  ab  +  cd  +  ac  +  bd  by  ab  +  cd  —  dc  —  bd. 


68  ELEMENTS   OF  ALGEBRA 

31.  x^^  —  x^y^ -{- x^'y*  —  xY' -\- y^  by  x^-\-y~. 

32.  -23?y  +  y^-^^x'y-^-x'^-2xi/  hy  x'^-\-2xy  +  f. 

33.  a^  -\- h^  -\- c^  —  he  —  ca  —  ab  by  a-\-h  +  c. 

34.  a;"+2  _|_  2  a;"+i  —  3  aj'*  —  1  by  a;  +  l. 

35.  —  ax^  +  3  aajy^  —  9  ay^  by  —  aa;  —  3  ay^. 

36.  —a^y  +  y^  -\-  x^y^  +  x*  —  xy^  by  ic  +  2/. 

37.  ia^  +  ia  +  i  by  ia-|. 

38.  |a^-2aj  +  f  by  \x^\. 

39.  f  a^  +  ic?/  4-  f  2/^  by  i  a;  -  1 2/- 

40.  Ja^_|a^-f  by  ^a;2+|a^-|. 

41.  I  a^  —  aaj  —  I  a^  by  I  a.-^  —  1^  aa?  +  I  a^ 

42.  laaj  +  fa.'^  +  ^a^  by  fa^  +  f  a;2_|^^ 

43.  3  a?"*  -  2  x"^-^  +  4  a^'^-^  by  2  aj'^  +  3  x"*"^  -  4  a;'"-^ 

44.  3  a;'*-^  +  a;'*-2  -  2  x^^-i  -  4  a;''  by  2  a^"-^  +  3  a;"-'*. 

45.  4  a V*  —  a^a^"  4- 5  a;"  by  a^a^^^-^  +  6  a;"-^ 

46.  3  a^'-'^y?  —  a"- V  +  a'*  by  aV'^  -  2  a;""^  -  3  ax''^"^. 

47.  4  a^"*+i  —  3  a^*"  —  2  aj'"+^  +  J  a^'""^ 

by   1  a^-+i  -  2  ar'^'+i  -  a^'"-^ 

48.  3(a4-?>)'-2(a4-&)'(aJ-2/)-4(a+6)(aj-2/)'+7(a;-2/)^ 

by  2{a^lSf{x-y)-Q>{a^h){x-y)\ 

81.  Removal  of  signs  of  grouping. 

Ex.  Remove  the  signs  of  grouping,  and  simplify, 


42_5[-12a;-3{-15x  +  3(8-7-3  (k)}]. 

The  expression  =  42  -  6  [-  12  x  -  3 {-  15  x  +  3  (3  ic  +  1)}] 
=  42  _  5  [-  12  X  -  3{-  6  a:  +  3}] 
=  42  -  6  [6  X  -  9] 
=  87-30  X. 


MULTIPLICATION  69 

Exercise  29. 
Eemove  the  signs  of  grouping,  and  simplify : 

1.  36-S5a-[6a  +  2(10-6)]S. 

2.  a  —  {h  —  c)  —  la-h  —  c  —  2\h-\-c\~\. 

3.  8(6  +  c)-[-Sa-6-3(c-6  +  a)J]. 

4.  2(36-5a)-7[a-6J2-(5a-6)j]. 

5.  Q>\a-2[b-^{c  +  d)']\-^a-^[b-4.iG-\-d)^]\. 

6.  5[a-2[a-2(a  +  a.')]|-4Ja-2[a-2(a  +  a;)"|5. 

7.  _l0Ja-6[a-(6-c)]5  +  60J6-(c  +  a)i. 

8.  _3J-2[-4(-a)]K5J-2[-2(-a)];. 

9.  _2S-l[-(x-2/)]S  +  S-2[-(a;-2/)]J. 

Multiply  together  the  following  expressions,  and  arrange 
each  product  in  descending  powers  of  x : 

10.  ax^-\-bx-\-l  and  ex +  2, 

11.  ax^  —  2bx-\-3c  and  x  —  1. 

12.  ic*  -h  aaj^  —  bx  —  c  and  ar'  —  a.^•^  —  6u;  +  c. 

13.  aa^  —  i»2 -f- 3  a;  —  6  and  oar*  +  a^ -f- 3  a;  +  i^. 

14.  x'^  —  €131?  —  bx^  -\- ex -{■  d  and  x^  4-  aar^  —  6a^  —  ca;  -f  d. 

82.   Multiplication  by  detached  coefficients. 

The  labor  of  multiplication  is  lessened  by  using  the  method 
of  detached  eoefficients  in  the  two  following  cases : 

(i)   When  two  polynomial  factors  contain  but  one  letter. 
Ex.  1.     Multiply  4x^-Sx^  +  2X-5  by  5 a;^  +  3 x  -  1. 
Writing  coefficients  only,  we  proceed  as  below  : 

4_    3+    2-    5 

6+    3-    4 


20  -  15  +  10  -  25 
+  12-    9+    6-15 

-  16  +  12  -    8  4-  20 
20  -    3  -  15  -    7-23  +  20 


70  ELEMENTS   OF  ALGEBRA 

Inserting  the  literal  factors,  whose  law  of   formation  is  seen  by 
inspection,  we  have  for  the  complete  product, 

20x^-Sx^-l6x^-7x'^-  23 x  +  20. 

(ii)   When  each  of  two  polynomial  factors  is  homogeneous 
and  contains  only  two  letters. 

Ex.  2.     Multiply  5  a*  +  4  a^^  _  3  ^fts  +  2  6*  by  a^-2  h\ 

5_l_4_}_0—    3  +  2  I^  th^  first  expression,  the  term  con- 

1  4.  0  —    2  taining  a^h^  is  lacking  ;  that  is,  its  co- 

efficient  is   zero,  which   is  written   in 

"^     "^       ~       "^  the  line  of  coefficients.     In  the  second 
~       ~       ~  expression,  the  term  containing  ah  is 


5  +  4  —  10  —  11  +  2  +  6  —  4     missing;  hence  its  coefficient  is  zero. 

In  the  method  of  detached  coeffi- 
cients, the  zero  coefficients  must  evidently  be  written  with  the  other 
coefficients. 

Inserting  the  literal  factors,  whose  law  of  formation  is  seen  by 
inspection,  we  have  for  the  complete  product, 

5  a6  +  4  a55  _  10  a*62  _  n  a%^  +  2  a'^h^  +  6  a&s  _  4  56. 

Observe  that  the  entire  number  of  coefficients  (zero  coefficients  being 
included)  in  the  product  is  one  less  than  the  number  of  coefficients  in 
both  the  multiplicand  and  multiplier  together. 

Exercise  30. 

1.  Multiply  ^J^2x^-x'^^x-l  by  x^-2x-3. 

2.  Multiply  3a^  +  2a2_5a-|.4  by  2a^-^a-2. 

3.  Multiply  a^  +  ^x'y-A.xif  +  ^Tf  by  2x^-3xhj  +  f, 

4.  Multiply  3  a^  -  2  a^6  -  4  a^h^  -  ab'  by  a^-2  h\ 

5.  Multiply  ^x''  -Z:^y  -^1  xy^  ^-2f  by  y?-\-'6f. 

6.  Rework  by  detached  coefficients  those  examples  in  ex- 
ercise 28,  from  19  to  42,  to  which  the  method  is  applicable. 


CHAPTER   VI 
DIVISION  OF  INTEGRAL  LITERAL  EXPRESSIONS 

83.  Division  is  the  inverse  of  multiplication.  Having  given 
a  product  and  one  factor,  division  is  the  operation  of  finding 
the  other  factor. 

That  is,  to  divide  one  number  by  another  is  to  find  a  third 
number  which  multiplied  by  the  second  number  gives  the 
first. 

Thus,         -  12  --       3  =  -  4  ;   for  -  4  X        3  =  -  12, 

and  _12h-(-3)=      4 ;    f or       4x(-3)  =  -12. 

As  in  Arithmetic,  the  given  product  is  called  the  dividend, 
the  given  factor  the  divisor,  and  the  required  factor  the 
quotient. 

84.  Law  of  Quality.  In  each  of  the  following  identities 
the  third  number  multiplied  by  the  second  gives  the  first; 
hence  by  definition  the  third  number  in  each  case  is  the 
quotient  of  the  first  divided  by  the  second. 

+(a6)-^-a=-6;     "(aft) --+« =-6.    J  ^^ 

From  identities  (1)  it  follows  that, 

The  quotient  is  positive  2vhen  the  dividend  and  the  divisor 
are  like  in  quality ;  and  negative  when  they  are  opposite  in 
quality. 

Tlie  arithmetic  value  of  the  quotient  is  equal  to  the  quotient 
of  the  arithmetic  value  of  the  dividend  by  that  of  the  divisor. 

71 


72  ELEMENTS   OF  ALGEBRA 

Any  number  divided  by  ^1  is  equal  to  the  number  itself. 
Any  number  divided  by  ~1  is  equal  to  its  arithmetically 
equal  opposite  number. 

Exercise  31. 
Perform  each  of  the  following  indicated  operations : 

1.  -25 -5.  5.    75 -(-25).  9.    21  -  (- 1). 

2.  36  ^(-6).  6.    -72 -(-6).  10.    -  36 -f- 4. 

3.  _5i^(_3).        7.    _  105 -(-21).       11.    -1^|. 

4.  _33^(_l).        8.    -144-24.  12.    l-^(-|). 

Find  the  value  of  (a?  +  ?/)  —  z, 

13.  When  x  =  —  15,  y  =  —  S,  z  =  6. 

14.  When  x  =  —  AS,  y  =  6,  z  =  — 
Find  the  value  of  (x  —  y)-r-  (a  4-  b), 

15.  When  05  =  22,  y  =  -2,  a  =  5,  &  =  3. 

16.  When  a?  =  -  21,  y  =  G,  a  =  -7,  b  =  6. 

85.  From  the  definition  of  division  we  have 

quotient  x  divisor  =  dividend. 

That  is,  since  the  quotient  of  N  divided  by  a  is  A^  —  a, 
we  have, 

(N^a)xa  =  N.  (1) 

86.  The   reciprocal   of   a  number   is   1   divided  by  that 
number. 

Since  their  product  is  +  1,  any  number  and  its  reciprocal 
have  the  same  quality. 

E.g.,  the  reciprocal  of  4  is  ^  ;  the  reciprocal  of  —  4  is  1  -4-(—  4) 
or  —  ^ ;  and  the  reciprocal  of  —  |  is  1  -4-  (—  |),  or  —  f. 


DIVISION  73 

87.  Dividing  by  any  number  except  zero  gives  the  sams 
result  as  multiplying  by  the  reciprocal  of  thai  number. 

That  is,  N-^a  =  Nx(l^a).  (1) 

Proof.  The  second  member  of  (1)  multiplied  by  a  is, 
by  §  ^5,  equal  to  N\  hence  it  is  the  quotient  of  N  divided 
by  a. 

Ex.  1.     16  H-  4  =  IG  X    ^  =  4. 

Ex.2.     16-(-4)=10  x(-i)  =  -4. 

88.  The  commutative  law  for  division. 

Ex.1.     _40^(-2)-(-5)  =  -40x(-^)x(-^)  =  -4.    (1) 

Ex.2.     ^-^(_|)^(-|)=^x(-3)x(-^)=f  (2) 

Since  we  can  change  the  order  of  the  factors  in  the  second  member 
of  either  (1)  or  (2),  we  can  also  change  the  order  of  the  divisors  in 
the  first  member  of  either  identity  ;  this  illustrates  that, 

The  commutative  law  holds  for  division  as  well  as  for 
multiplication,  provided  the  sign  of  operation,  -r-  or  x,  before 
each  number  is  transferred  with  the  number  itself. 

That  is,              N  xb^c  =  N^cxb.  (1) 

Proof                  Nxb-^c  =  Nxbx(l-hc)  §  87 

=  Nx{l^c)xb  §49 

=  N-r-  cxb.  §  87 

Ex.     (-60)x(-22)-f-(-  l5)  =  (~60)-(-  15)x(-22) 

=  4  x(-22)  =  -88. 

89.  A  product  of  two  or  more  factors  is  divided  by  a  num- 
ber if  any  one  of  the  factors  is  divided  by  that  number. 

Proof  {ab)-^c  =  a^cxb={a^c)b,  §88 

or  {ab)  -i-  c  =  b  -T-  c  X  a  =  (b  ^  G)a.  §  88 


74  ELEMENTS   OF  ALGEBRA 

90.  Any  indicated  quotient  is  called  a  fraction. 

A  quotient  is  often  indicated  by  placing  the  dividend 
over  the  divisor  with  a  line  between  them. 

E.g.^  a  -^  b,  -,  and  a/b  are  but  different  ways  of  indicating  that 
b 
a  is  to  be  divided  by  b. 

Each  of  these  expressions  is  a  fraction,  a  being  the  dividend,  and 
b  the  divisor.  The  dividend  and  divisor  of  a  fraction  are  often  called 
its  numerator  and  denominator  respectively. 

When  the  dividend  or  divisor  consists  of  more  than  one 
term,  the  horizontal  dividing  line  in  a  fraction  serves  as  a 
sign  both  of  division  and  of  grouping. 

E.g.,  in  the  fraction  ^  ~      the  horizontal  dividing  line  takes  the 
c  +  d 
place  of  both  the  sign  of  division  and  the  two  parentheses  in  the  form 

(a-6)^(c  +  c?),  or  (a-  6)/(c  +  d). 

In  §  1  any  fractional  number  as  5/6  was  regarded  as 
(1/6)  X  5 ;  but  it  can  also  be  regarded  as  5  -r-  6 ;  for 

]Sr-r-a  =  Nx(l-^a)  =  (1/a)  x  JV.  §§  87,  49 

91.  The  product  of  two  or  more  fractions  is  equal  to  the 
product  of  their  dividends  divided  by  the  product  of  their 
divisors;  and  conversely. 

mi-  J.  •  a     b     c      abc  ,^. 

That  IS,  _._._  = (1) 

X    y     z      xyz  ^  ^ 

^       .     a     b    c  a  b  c  ..„ 

Proof.     -'~'-'X-y-z  =  ~'X'-'y'-'Z  §49 

'^      X    y     z  ^  X  y     ^     z 

=  abc.  §  85 

Dividing  each  member  by  xyz,  we  obtain  (1). 

Ex        4    ^     3    ^-2_ -(4x3x2)^  8 
_  5      _  7      _  3      -(5x7x3)     35 


DIVISION  75 

92.   Quotient  of  powers  of  the  same  base. 

Ex.  1.  a^  ^  a-  =  a^-'^  =  a^  ;  for  a^  x  a'^  =  w'. 
Ex.  2.  a?  -^  a^  =  o?-^  =  a*  ;  for  a'^  x  a^  =  a?. 
These  examples  illustrate  the  following  law  : 

If  /n  >  /I,  the  quotient  of  the  mth  power  of  any  base  divided 
by  the  nth  power  of  the  same  base  is  equal  to  the  (m  —  n)th 
power  of  that  base;  and  convex 


That  is,  a'"  -T-  a"  =  a'"'". 

Proof  a"*-"  X  a"  =  a*"-''+''  =  a"*.  §  76,  83 


Ex.  1.     Divide  20a*6S  by  -bah^. 


91 


20q^ft5_  20     a^    6^ 
—  6  ab^~  —  5     a     6* 

=  -  4  a362.  §§  84,  92 


91 


Ex.  2.     Divide  -  ba^H^  by  lla262a;2. 

-ba%^x^_-b    a^    b^    7? 
11  a262a;-^  ~  11    '  d^'  b'i'  ic2 

=-/x  •  1  .  62  .x  =  -^bH. 
These  examples  illustrate  the  following  section. 

93.  The  quotient  of  one  monomial  by  another.  By  the  con- 
verse of  §  91  we  have  the  following  rule : 

Divide  the  numeral  factor  of  the  dividend  by  that  of  the 
divisor,  observing  the  law  of  quality;  after  this  write  the  quo- 
tient of  their  literal  factors,  obset^ving  the  laiv  of  exponents. 

Ex.  1.     -  84  a^xs  -f-  12  a*x=-  7  ax"^. 

Ex.  2.     77  a^x^y^  ^  (  -  7  ax^y)  =-11  axyK 

Check.  Multiplying  the  obtained  quotient  by  the  given  divisor,  we 
obtain  the  dividend  ;  hence,  the  division  is  correct. 


T6  ELEMENTS   OF  ALGEBRA 

Exercise  32. 
Divide : 

1.  -  72  a^  by  -  9  a.  6.  84  affz'  by  -  7  icyV. 

2.  84a3by-7al  7.  28  a^d^  ^y  _  4  ^3^^ 

3.  -  35  a^  by  7  ic^.             .  8.  -  35  a%^  by  5  ab. 

4.  4a^6V  by  —ab^c\  9.  —  16a^/  by  —  4ic/. 

5.  -  12  a^ft'^c^  by  -Sa^ftc^.  10.  36  m%^2  ^^y  9  ^6^9^ 

11.  96a*afz*  by  12  aV;^^ 

12.  -  256  xyz^'  by  -  8  a;y;2«. 

13.  SAaWc'  by  14a&V.  16.    - 144  aV  by  -24aV. 

14.  -  16  %a;2  by  -  2  a;?/.  17.    -  3  x'^+^  hj  5  x'^+\ 

15.  50  yV  by  —Ba^y.  18.    _  4  a7'«+«2/'»+"  by  7  x*"?/"*. 

19.  5  a;"+V'+^  by  -  8  aj^^/"*- 

20.  -  7  a;^+'^"'+2  by  -  2  a;"- V"^- 

21.  —  42  x«+3a"*-i  by  —  7  a^'-^a'^-l 

22.  —  50  a7"+«?/'"+*  by  25  a;"-*^"*-". 

94.  Distributive  law  for  division.  The  quotient  of  one  ex- 
pression divided  by  another  is  equal  to  the  sum  of  the  results 
obtained  by  dividing  the  parts  of  the  first  expression  by  the 
second;  and  conversely. 

That  is,        «±At£_±^^«_l_i  +  £+..,  (o 

X  XXX 

Principle  (C)  lies  at  the  basis  of  division  in  Arithmetic;  e.g.,  to 
divide  894  by  6  we  separate  894  into  the  parts  600,  240,  and  54,  divide 
each  of  these  parts  by  6,  and  add  the  results. 

Thus  8M^0Og^24O^54^j00^,„^9^j^g_ 

6         6         6        6 


DIVISION  77 


Proof.  ±±±±SL±^=(a-^b  +  c+:.)^  §87 

X  X 

=  a-  +  /)-  +  ci+...  §60 


:-  +  -+-  +  -.  §87 


95.   To  divide  a  polynomial  by  a  monomial. 
By  the  distributive  laAv  for  division,  in  §  94,  we  have  the 
following  rule : 

Divide  each  term  of  the  polynomial  by  the  monomial,  and 
add  the  resulting  quotients. 

Ex.  1.   Divide  l2x^-6ax^-2  a^x  by  3  x. 

3a;  ~  3x  3a;  3x  ^ 

=  4a;2- 5rta;-|a2. 

Ex.  2.   Divide  12  a"  +  0  a^  _  6  gS  by  -  3  a2. 

12q«  +  9a*-6a5_  12ff«         9  a*        -6a^  „q. 

=  -  4  a  -  3  a2  +  2  a8. 


Exercise  33. 
Divide : 

1.  5x^-7  ax-i-4:X  by  X.  6.  -24  a^-32  a;^  by  -Sir^. 

2.  a;'''-7ar^  +  4aj*by  a.-^.  7.  a^-a-ft-a-ft^  by  al 

3.  lOx^—Safi-^Sx*  by  a;^.  8.  a^  — a6-ac  by  —a. 

4.  27  3^-363.*^  by  9ar\  9.  .r^-a^-oa;  by  -a;. 

5.  15  ar' - 25 a;*  by  -5. T^.  10.  3a;«-9.Tyby  -3aj. 

11.  4a^6^~8a%^  +  Ga6^  by  -2a6. 

12.  —  3a^  +  f  a^  — (j^2;  by  —  f  a;. 

13.  _5a^  +  |.^y  +  j^0a,    by  -fa;. 

14.  ia.V-3.r^y-5.Ty  by  -^sc^y^ 


78  ELEMENTS   OF  ALGEBRA 

15.  J  a^x  —  Jg-  abx  —  |  acx  by  |  ax. 

16.  -  2  a^or^  + 1  aV  by  |  a^x. 

17.  25(aj  +  2/)'-3a(a;  +  ?/)2  +  106(a^  +  ?/)  by  5(aj  +  2/). 

18.  -S(a-by -12  x(a -by- 16  y  (a -by  by  4(a-6)2. 

19.  Ga^'"  — 4  a-'"  by  2  a". 

20.  10  2/"+V  - 15  r^'^'  by  -  5  y^z. 

Divide  12  a;2«+y  - 16  x^'^+Y  -  20  aj^^+y  by : 

21.    4  a;".  22.    —  «'»+y. 

23.    —  Sa^^n^^.  24.    ia;2n-3^2^ 

25.  Divide  4  a;-'^+y  —  16  a;-"+V +i  by  4aj2«2/«. 

26.  Divide  - 15  a^+Y+^  +  21  x''+Y+'^  by  3  a;*+y+2, 

96.   To  divide  one  polynomial  by  another. 

Let  it  be  required  to  divide 

2  x^y"^  —  x^y  +  x^  —  xy^  +  y^  by  y"^  —  xy  +  x^. 

First  arrange  dividend  and  divisor  in  descending  powers  of  «,  for 
convenience  placing  the  divisor  to  the  right  of  the  dividend  as  below  : 


a*  —  x^y  +  2  x'^y^  —  xy'^  +  y* 
x^  —  x^y  +     x'^y'^ 


x^  —  xy  +  2/2  Divisor 


x^  +  y'^  Quotient 


x^y'^  —  xy^  +  2/* 
x^y^  —  xy^  +  y* 


From  the  law  of  exponents  we  know  that  a;*,  the  term  of  the  highest 
degree  in  x  in  the  dividend,  is  the  product  of  the  terms  of  highest 
degree  in  the  divisor  and  the  quotient ;  hence,  the  first  term  of  the 
quotient  is  x^  -r-  x^,  or  x"^.  Multiply  the  divisor  by  x^  and  subtract  the 
result  from  the  dividend. 

The  remainder,  xV  -  xy^  +  y'^,  is  the  product  of  the  divisor  by  the 
other  terms  of  the  quotient ;  hence,  x^ij'^,  the  first  term  of  the  remain- 
der, is  the  product  of  the  first  term  of  the  divisor  and  the  second 
term  of  the  quotient.  Therefore  the  second  term  of  the  quotient  is 
a;2y2  ^  a;2^  or  y"^.  Multiplying  the  divisor  by  y^  and  subtracting  the 
result  from  x^y^  —  xy^  +  ?/*,  we  have  no  remainder. 

Hence  the  required  quotient  is  x^  +  y\ 


DIVISION  79 

Observe  that  by  the  above  process  the  dividend  was  separated  into 
the  two  parts  ar*  —  x^y  +  x^y'^  and  x'^y^  —  xy^  +  y^  ;  hence,  by  the  distrib- 
utive law  for  division,  we  have 

x^  —  x^y  +  2  x-y'^  —  xy^  +  y*  _x*  —  x^y  +  x-y'^     x^y"^  —  xy^  +  y^ 


x/^  —  xy  +  «/2  x^  —  xy  +  y^         x^  —  xy  -{■  y^ 

=  x2  +  y2. 

If  the  dividend  and  divisor  were  arranged  in  ascending  powers  of  x, 
the  quotient  would  be  obtained  in  the  form  y^  +  x^. 

Hence,  to  divide  one  polynomial  by  another,  we  have  the 
following  rule : 

Arrange  the  dividend  and  divisor  in  descending  powers  of 
some  common  letter. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  and  write  the  result  as  the  first  term  of  the  quotient. 

Multiply  the  divisor  by  this  first  term  of  the  quotient,  and 
subtract  the  resulting  pj-oduct  from  the  dividend. 

Divide  the  first  term  of  the  remainder  by  the  first  teiin  of 
the  divisor,  and  write  the  result  as  the  second  term  of  the 
quotient. 

Multiply  the  divisor  by  this  second  term  of  the  quotient,  and 
subtract  the  resulting  product  from  the  remainder  previously 
obtained. 

Treat  the  second  remainder,  if  any,  as  a  new  dividend  and 
go  on  repeating  the  process  iintil  the  remainder  is  zero,  or  is 
of  a  loiver  degree  in  the  letter  of  arrangement  than  the  divisor. 

Ex.  1.  Divide  2  a  -  4  a^  +  3  a^  -  1  by  1  -  a. 

Arranging  dividend  and  divisor  in  descending  powers  of  a,  we  have 


3a8-4a2H-2a- 

-1 

-a  +  1 

3a'-3a2 

-  a2  +  2  a 

-  a2-|-     a 

-  3  a-2  +  a  - 

a  - 
a  - 

-  1 
-1 

80 


ELEMENTS   OF  ALGEBRA 


Ex.  2.    Divide  x^y'^  +  x^  +  y^  by  y"-  —  xy  +  x'^. 

Arranging  dividend  and  divisor  in  descending  powers  of  x,  we  have 


ic*  +  x-y'^ 

xl^  —  x^y  +  xhf' 

x^y 

x^y  —  x-y'^  +  xy^ 


+  2/* 


X?/  +  y2 


yp-  +  xy  ^ 


xhf-  —  xy^  +  y^ 


Ex.  3.    Divide  16  a*  -  1  by  2  a  -  1. 


16  a4 

-1 

16  a4. 

8^3 

8a3_ 

-4a2 
4a^ 

4a2_ 

-2a 
2a-l 
2rt-l 

2a 


8  rt3  4  4  ^2  +  2  a  +  1 


Exercise  34. 
Divide : 

1^    a;2  +  3a^  +  2  by  a;  + 1.  4.    3a;2  +  10a;  +  3  by  a;  +  3. 

2.  a2_ii^_^3o  by  a-h.         5.    oa^  +  lla;  +  2  by  x^2. 

3.  aj2_7^_^i2  by  a;-3.  6.    5a^  +  16a;H-3  by  ic  +  3. 

7.  2x2-t-llx+5  by  2a.'-}- 1. 

8.  2a^H-17a;-f  21  by  2ic-h3. 

9.  4a)2-f-23ic  +  15  by  4aj  +  3. 

10.  6a^-7aj-3  by  2a;-3. 

11.  12a2-7aa;-12a.'2  by  3a-4a;. 

12.  15  a^  -h  17  aic  —  4  a^  by  3  a  -f-  4  a;. 

13.  12  a2  _  11  ac  -  36  c^  by  4  a  -  9  c. 

14.  60a;2-4ic?/-45?/2  by  lOic-Oi/. 


DIVISION  81 

15.  -A.xy-loy-^-^ijx'  hy  12x-6y. 

16.  10()a:3-3.T-13a;2  ^^^  3-|_25x. 

17.  16-96ic4-216ic2-216a:3  +  8i«^  by  2-3a;. 

18.  x'-x'-'^x-nhy  x'-\-^x-\-^. 

19.  22/3_32^2_gy_l  by  22/--52/-1. 

20.  Q>m^-m'^-l^m-\-S  by  3m2H-4m-l. 

21.  6a'-13a^  +  4a3  +  3a2  by  ^a^-2o?-a. 

22.  a.'*  +  .T3  +  7iv2-6a;  +  8  by  x'  +  2x-[-^. 

23.  a^-a3-6a-  +  15a-9  by  a-4-2a-3. 

24.  a^  +  Oa^ H-13a2  +  12a4-4  by  a2  4-3a  +  2. 

25.  2a;^-ar»4-4a^  +  4a;-3  by  ar^-a;  +  3. 

26.  a;5-5a;^  +  9ii-3-6a.-2-a;  +  2  by  a^-3a;  +  2. 

27.  ar'-4a;*4-3it'3-|.3a.-2_3a'  +  2  by  a^-a;-2. 

28.  30a-*4-lla.'3-82x2_i2a;  +  48  by  2.T-4  +  3a^. 

29.  69?/-18-71 2/3  +  28^-352/2  ^^  4^y2_x^yj^Q^ 

30.  0^--15^•^  +  4^^  +  7fc2-7A:  +  2  by  3A:3-A:  +  1. 

31.  2x^-^x  +  x^-\-12-lx'  hy  x'  +  2-3x. 

32.  a^-2a;^-7x'3  +  19x2_i0^.  1,^  ^_r'^_j_5 

33.  Ux^  +  ^^a?y  +  l^^f-{-A^xi/-{-Uy'   by  20.-2  +  50^2/ 
+  72/'. 

34.  x^ -{- x^y - a?f  -\- Qi? -2 xy'^ -\-  f  by  a^  +  a^-/. 

35.  x^  -2y^*  -7  afy*  -7  xy^'-  -\-Uxy  by  x-2y\ 

36.  a^  +  63  _^  c^  _  3  ((6c  by  a  +  6  +  c. 


82 


ELEMENTS   OF  ALGEBRA 

a3  _  3  a6c  +  &^  +  c3    a  +  h  +  c 

aS  +     a'^b  +  a^c 

a'^  -  ab  ~  ac  +  b'^  -  be -\-  c^ 

-  a%  -  a2c  -  3  abc  +  b^ -\- c^ 
-a^b-ab^-    abc 

-  a^c  +  a62  _  2  abc  +  b^  +  c^ 

—  a^c            —     abc  —  ac^ 

ab^  -  abc  +  ac^  +  b» -\- c^ 
ab-^                      +  b^  +  b^c 

-  abc  +  ac2  -  6%  +  c- 

-  abc            -  b'^c  -  bc^ 

ac2+  6c2  +  c3 
ac2  +  ?)C2  +  c^ 

Here  we  arranged  the  dividend  and  divisor  in  descending  powers  of 
a,  and  gave  b  precedence  to  c  throughout. 

37.  a^ -\- if  —  z^ -\- 3  xyz  hj  x-\-y  —  z. 

38.  S  x^  —  y^ -{- z^ -{- 6  xyz  by  y  —  z~2x. 

39.  27a^-Sb^-^(^-\-lSahchj  3a-2b-\-c 

40.  a^  +  3a26  +  3ai!>2  +  63  +  c3  by  a  +  &  +  c. 

41.  a' -\-b'  +  c' -2b'<^ -2 a'c" -2a'b'  by  a  +  ^  +  c. 

42.  ^:^-^-^^^xf^  +  j\fhjix-^ly. 

43.  |a3_  9^2^_^_2_7^^_27ar^  by  ia-3a;. 

44.  ^aS-^i^a^  +  ^-Va-eV  by  ^a-i- 

45.  I^V H-^a^  by  ia^  +  iac. 

46.  ^9^a4_|a^-7a2  +  .|a4.i_§  by  f a^-f-a. 

47.  36a52+i2/'  +  i-4a^-6aj  +  i2/  by  Gic-iy-i 

48.  ^8_o^5_2  4|ci,a;4by  |a-fa;. 

49.  43  a^'"-^  +  6  «2-+i  _  29  aj^-  -  20  aj^--^  by  2  a;'"  -  5  x'^-\ 

6  aj^'w+i  -  29  x-^  +  43  a;2'«-i  -  20  x^""-^  |  2  a;'"  -  5  a;"*-^ 

6  a:2"*+i  -  15  x^"* 3  x'^+^  -  7  a;"*  +  4  x""-^ 

-  14  x2«  +  43  ic2'»-i 

-  14  a;2»»  +  35  a;2'»-i 


8  a;2»»-i  —  20  a;2»*-2 
8  a.2m-i  _  20  x^""-^ 


DIVISION  83 

50.  ic'*"  —  a^"^/"'  +  x'^if"^  —  y^""  by  x^  —  2/"*. 

51.  6a3'»-25a-«  +  27a''-o  by  2  a'* -5. 

52.  6a'"'-lla^  +  13a2«  +  23a»"H-2-3a"  by  3a"4-2. 

53.  12  a;"+i  +  8  a?"  -  45  a;"-^  +  25  x''-'-  by  6  a  -  5. 

54.  i^ia'  -  Li3.ab  -{-9  ac  +  2b'  -  be  by  Ja-36+fc. 

55.  (5  +  c) a^  —  bcx  -\- x^  —  be (b -\- c)  by  a^^  —  6c. 

56.  a.'^  +  (a  +  6  +  c)  ar  4-  (a6  +  ac  +  6c)a;  +  a6c  by  x-\-b. 

57.  a?^  4-  (a  H-  6  —  c) a^  +  (a6  —  ac  —  6c) a;  —  a6c  by  x—  e. 

97.  When,  as  in  each  example  given  above,  the  division 
is  exact,  the  quotient  is  the  same  whether  the  dividend 
and  divisor  are  arranged  in  deseending  or  in  aseending 
powers  of  any  common  letter.  But  when  the  division  is 
not  exact,  the  partial  quotient  obtained  with  one  arrange- 
ment is  not  the  same  as  that  obtained  with  the  other. 

E.g.,  ^l±l  =  x-l  +  -^;  (1) 

while  1+^^=1 -a; +  2x2 -2x8+^^.  (2) 

1  +  x  1+x  ^  ^ 

Here  the  partial  quotients  x  —  1  and  1  —  x  +  2  x^  —  2  x^  are  evi- 
dently unequal.  The  entire  quotients,  or  the  second  membei-s  of  (1) 
and  (2),  are,  of  course,  identical. 

In  (1)  the  remainder  is  of  a  lower  degree  than  the  divisor. 

In  (2)  the  division  can  be  carried  to  any  number  of  terms. 

When  arranged  in  ascending  powers  of  some  common 
letter,  an  expression  of  a  lower  degree  can  be  divided  by 
one  of  a  higher  degree  in  the  letter  of  arrangement. 

E.g.,  -l_  =  l  +  a;  +  aj2  +  aj3  +  ...+aj"-i+--^^. 
1  —  aj  1  —  X 


84  ELEMENTS   OF  ALaEBRA 

Exercise  35, 
Divide : 

1.  a^ -^  y^  hj  X -\- y.  5.  x^  —  a^  hj  x -\- a. 

2.  x^  -{-  if  hj  X  —  y.  6.  1  by  1  +  ic  to  4  terms. 

3.  x^  —  y^  bj  X -\- y.  7.  1  +  a;  by  1  +  ic^  to  5  terms. 

4.  x^  +  y*hj  x-}-y.  8.  1 -f- 2  a?  by  1  —  3  a;  to  4  terms. 

9.  Divide  2  by  1  -}-  a;  and  thus  reduce  the  second  member 
of  identity  (1)  in  §  97  to  the  form  of  the  second  member 
of  (2). 

98.  Zero  divided  by  any  number,  except  zero,  is  equal  to 

zero. 

That  is,  when  a^O,  0^a  =  Q.  §§  85,  74 

Conversely,  if  a  quotient  is  zero,  the  dividend  is  zero. 

Zero  as  divisor  will  be  considered  in  Chapter  XXVII. 
Prior  to  that  chapter  it  will  be  assumed  that  any  expression 
used  as  a  divisor  does  not  denote  zero. 

99.  Division  by  detached  coefficients.  In  §  82  we  considered 
two  cases  in  which  the  work  of  multiplication  could  be  shortened  by 
using  the  method  of  detached  coefficients.  In  the  same  two  cases  the 
labor  of  division  can  be  lessened  by  using  detached  coefficients  and  an 
arrangement  of  terms  known  as  Horner's  method  of  synthetic  division. 
This  method  is  illustrated  by  the  following  examples  : 

Ex.  1.     Divide  2x^-1  x^-\-2x^- x^-Qx-^ 20  by  2x3-3x2+4aj-5. 

cS   i      3 

S  "      5 

Quotient  1-2-4J         0  0  0  Remainder 

Inserting  in  the  quotient  the  literal  factors,  whose  law  of  formation 
Is  seen  by  inspection,  we  have  for  the  complete  quotient  x'^  —  2x  —  A. 

Explanation.  The  modified  divisor,  or  the  column  of  figures  to  the 
left  of  the  vertical  line,  consists  of  the  coefficients  of  the  divisor,  the 


2 

-7 

+  2 

-    1 

-    6 

+  20 

3 

-4 

5 

-6 

+    8 

-10 

-12 

+  16 

-20 

DIVISION  85 

quality  of  each  coefficient  after  the  first  being  changed;  this  change 
of  quality  enables  us  to  replace  the  operation  of  subtraction  by  that  of 
addition  at  each  successive  stage  of  the  work. 

Observe  that  the  number  of  coefficients  in  the  quotient  will  be  one 
more  than  the  number  of  coefficients  in  the  dividend  minus  the  num- 
ber of  coefficients  in  the  divisor,  in  this  case  1  +  6  —  4,  or  3  (§  82). 
Thus,  the  numbers  to  the  left  of  the  vertical  bar  are  the  coefficients  of 
the  quotient^  and  those  to  the  right  of  this  bar  are  the  coefficients 
of  the  remainder. 

Dividing  the  first  coefficient  of  the  dividend  by  the  first  coefficient 
of  the  divisor,  we  obtain  the  first  coefficient,  1,  of  the  quotient.  Multi- 
plying the  modified  coefficients  of  the  divisor  (3,  —  4,  5)  by  this  first 
coefficient  of  the  quotient,  we  obtain  line  (1). 

Adding  the  coefficients  in  the  second  column  to  the  right  of  the 
divisor,  and  dividing  the  sum  by  the  first  coefficient,  2,  of  the  divisor, 
we  obtain  —  2,  which  is  the  second  coefficient  of  the  quotient.  Multi- 
plying the  modified  coefficients  of  the  divisor  by  this  second  coefficient 
of  the  quotient,  we  obtain  line  (2). 

Adding  the  coefficients  in  the  third  column  and  dividing  the  sum 
by  the  first  coefficient  of  the  divisor,  we  obtain  the  thirds  or  last, 
coefficient  of  the  quotient.  Multiplying  the  modified  coefficients  of 
the  divisor  by  this  third  coefficient  of  the  quotient,  we  obtain  line  (3). 

Lines  (1),  (2),  and  (3)  are  evidently  the  coefficients  of  the  three 
partial  products  obtained  by  multiplying  the  divisor  by  each  term  of 
the  quotient,  the  first  term  of  each  product  being  omitted  and  the 
quality  of  the  others  being  changed. 

Hence  by  adding  each  of  the  vertical  columns  after  the  third,  we 
obtain  the  coefficients  of  the  remainder. 

Here  the  coefficients  are  all  zero,  and  the  division  is  exact. 

Ex.  2.     Divide  2x^-1 7^y+\2  v^y"^  -  8  x^y^  ■\-  a; V  by  2  x^  -  3  x^y  -  y^. 


2 

2 

-7     +12 

-8 

+  1 

+  0 

+  0 

(1) 

3 

+  3          0 

+  1 

(2) 

0 

-    6 

0 

-2 

(3) 

1 

+  9 

0 

4-3 

(4) 

+  3 

0 

+  1 

(5) 

1      _2     +    3     +1  I  +2     +3     +1 

Inserting  the  literal  factors,  we  have  for  the  quotient  x^  —  2  x'^y 
+  3  xy"^  +  ?/'^,  and  for  the  remainder  2  x'^^/*  +  3  a;?/^  +  y^. 

Explanation.  The  terms  in  xy^  and  y^  are  missing  in  the  dividend, 
and  the  term  in  xy"^  in  the  divisor;   hence  their  zero  coefficients  are 


86  ELEMENTS   OF  ALGEBRA 

written  with  the  other  coefficients.     The  sums  of  the  vertical  columns 
after  the  fourth  give  the  coefficients  of  the  remainder. 

To  find  the  remainder  after  one  term  of  the  quotient,  add  lines  (1) 
and  (2)  after  the  first  vertical  column  ;  to  find  the  remainder  after 
two  terms  of  the  quotient,  add  lines  (1),  (2),  and  (3)  after  the  first 
two  vertical  columns  ;  to  find  the  remainder  after  three  terms  of  the 
quotient,  add  lines  (1),  (2),  (3),  and  (4)  after  the  first  three  vertical 
columns. 

Exercise  86. 

Divide : 

1.    a-^  -  4  or'  +  2  a;2  +  4  a;  4- 1  by  ^  -  2  a;  -  1. 

3.    x^^-{-x^-\-l  by  0^  +  .1^  +  1. 

by  2x^-3xy-\-Ay^ 

5.  Sf-22xy'-^20a^rf^xy-7x'y  +  6x^    . 

by  4:y^  —  3xy  -^2x^0 

6.  a' -3 a'b^  +  8  ab^  -  5b^  hj  a^ -4.ab -{-b'  to  four  terms 
in  the  quotient. 


CHAPTER  VII 
INTEGRAL  LINEAR  EQUATIONS   IN  ONE   UNKNOWN 

100.  An  integral  equation  is  an  equation  all  of  whose  terms 
are  integral  in  the  unknown.     (Review  §§  10,  16,  17.) 

E.g.^  2  a;2  +  3  =  2  X  and  ^  -}-  —  =  x  +  2  are  integral  equations. 
2        0 

101.  The  degree  of  an  integral  equation  in  one  unknown 
is  the  degree  of  its  term  of  highest  degree  in  the  unknown. 

A  linear  equation  is  an  equation  of  the^rs^  degree. 
A  quadratic  equation  is  an  equation  of  the  second  degree. 
A  higher  equation  is  an  equation  of  a  higher  degree  than 
the  second. 

E.g.,  3 ic  +  1  =  4  and  ax  +  6  =  0  are  linear  equations  in  x. 

6  x2  —  7  X  =  1  and  ax^  +  6x  +  c  =  0  are  quadratic  equations  in  x. 

6  x^  —  4  x^  +  3  X  4-  4  =  0  is  a  higher  equation  in  x. 

102.  A  root,  or  solution,  of  an  equation  in  one  unknown  is 
any  value  of  the  wiknown  ;  that  is,  it  is  any  number  which 
when  substituted  for  the  unknown  renders  the  equation  an 
identity. 

E.g.,  12  is  a  root  of  the  equation 

2  X  -  5  =  X  +  7. 

For,  putting  12  for  x  in  the  equation,  we  obtain  the  identity 

24  -  5  =  12  +  7. 

Any  root  of  an  equation,  since  it  satisfies  the  condition 
expressed  by  the  equation,  is  said  to  satisfy  the  equation. 

87 


88  ELEMENTS   OF  ALGEBBA 

103.    To  solve  an  equation  in  one  unknown  is  to  find  all 
its  roots.     In  solving  equations  we  use  the  principles  of 


EQUIVALENT  EQUATIONS. 

104.  Two  equations  in  one  unknown  are  said  to  be  equiv- 
alent, when  every  root  of  the  first  is  a  root  of  the  second, 
and  every  root  of  the  second  is  a  root  of  the  first. 

E.g.^  the  equations 

4a;-8  =  2-x  (1) 

and  bx=  10  (2) 

have  the  same  root,  i.e.,  are  equivalent ;  for  2  is  a  root  of  each  equa- 
tion, and,  as  will  be  seen  later,  2  is  the  only  root  of  either. 

In  solving  equations  we  need  to  know  what  operations  on 
the  members  of  an  equation  will  make  the  derived  equation 
have  the  same  root,  or  roots,  as  the  given  one. 

Of  such  operations  the  most  elementary  and  important 
are  found  in  §§  105,  106,  108,  109. 

105.  Identical  expressions. 
If  in  the  equation 

4(x-l)-(3x-2)=3,  (1) 

we  substitute  for  the  first  member  the  identical  expression  a;  —  2,  we 
obtain  the  equivalent  equation 

X  -  2  =  3.  (2) 

For,  as  is  easily  shown,  5  is  a  root  of  either  equation  ;  and,  as  will 
be  seen  later,  5  is  the  only  root  of  either. 

This  example  illustrates  the  following  principle  : 

If ,  for  any  expression  in  an  equation,  an  identical  expres- 
sion is  substituted,  the  derived  equation  will  he  equivalent  to 
the  given  one. 


INTEGRAL  LINEAR  EQUATIONS  89 

That  is,  ii  A  =  B  denotes  any  equation  in  one  unknown, 
as  X,  and  A  =  A'\  then  the  equations 

A  =  B  (1) 

and  A'  =  B  (2) 

have  the  same  root,  or  roots. 

Proof.  To  prove  that  equations  (1)  and  (2)  have  the  same 
root,  or  roots,  we  must  prove  that  every  root  of  (1)  is  a  root 
of  (2)  ;  and  conversely  that  every  root  of  (2)  is  a  root  of  (1). 

Since  A  and  A^  are  identical  expressions,  any  value  of  x 
which  when  substituted  for  x  will  make  either  one  identical 
with  By  will  make  the  othei?  identical  with  B  (§  32). 
Hence,  any  root  of  (1)  is  a  root  of  (2),  (§  102) ;  and  con- 
versely any  root  of  (2)  is  a  root  of  (1) ;  that  is,  equations 
(1)  and  (2)  have  the  same  roots,  i.e.y  are  equivalent. 

E.g.,  since,       3(a;  -  1)  -  {3  x  -  (2  +  x)}  =  x  -  1  ; 

the  equations        3(x  -  1)  -  {3  x  -  (2  +  x)}  =  5  (1) 

and  x-l  =  5  (2) 

have  the  same  root ;  that  is,  we  neither  lose  nor  introduce  a  root  by- 
substituting  for  3(x  —  1)  —  {3  X  —  (2  +  x)}  in  equation  (1)  its  identical 
expression  x  —  1. 

106.   Addition  or  Subtraction. 

If  to  both  members  of  the  equation 

2x-8  =  7-8  (1) 

we  add  8  +  x,  we  obtain  the  equivalent  equation 

3x  =  15. 

For,  as  is  easily  shown,  5  is  a  root  of  each  equation ;  and,  as  will 
be  seen  later,  5  is  the  only  root  of  either  equation. 
This  example  illustrates  the  following  principle  : 

If  identical  expressiojis  are  added  to,  or  subtracted  from, 
both  members  of  an  equation,  the  derived  equation  will  be 
equivalent  to  the  given  one. 


90  ELEMENTS   OF  ALGEBRA 

That  is,  if  31=  M',  the  equations 

A=B  (1) 

and  A±M=B±M'  (2) 

have  the  same  root  or  roots. 

Proof.     Any  root  of  (1)  makes  A  =  B.  §  102 

But,  when  A  =  B,      A±M=B±M'.  §  32,  (iii) 

Hence,  any  root  of  (1)  is  a  root  of  (2). 

Conversely,  any  root  of  (2)  makes  A  ±  M=  B  ±  M'. 

B\xt,when  A  ±M=B±M',      A  =  B.  §  32,  (iii) 

Hence,  any  root  of  (2)  is  a  root  of  (1). 

Therefore,  equations  (1)  and  (2)  are  equivalent.         §  104 

If,  to  each  member  of  the  equation 

ax  —  b  =  ex  —  d,  (1) 

we  add  —  ex  and  +  6,  we  obtain  the  equivalent  equation 

ax  —  ex  =  b  —  d.  (2) 

Adding  —  ex  to  both  members  of  equation  (1)  removes  the  term 
+  ex  from  the  second  member,  and  transfers  it,  with  its  sign  changed 
from  +  to  — ,  to  the  first  member.  Likewise,  adding  +  6  to  both 
members  of  (1)  removes  the  term  —  b  froili  the  first  member,  and 
transfers  it,  with  its  sign  changed  from  —  to  +,  to  the  second  mem- 
ber. This  example  illustrates  the  following  important  application  of 
the  principle  proved  above. 

If  any  term  is  transposed  from  one  member  of  an  equation 
to  the  other,  its  sign  being  changed  from  -\-  to  —,  or  from  — 
to  +,  the  derived  equation  has  the  same  root  or  roots  as  the 
given  one. 

107.  An  expression  is  said  to  be  unknown,  or  knoivn,  ac- 
cording as  it  does,  or  does  not,  contain  an  unknown  number. 

E.g.,  if  X  is  an  unknown  number,  a:  —  2  is  an  unknown  expression  ; 
if  a  is  a  known  number,  9  +  5  a  is  a  known  expression. 


INTEGRAL  LINEAR  EQUATIONS  91 

108.   Multiplication. 

If  both  members  of  the  equation 

2      4     3      12  ^^ 

are  multiplied  by  12,  we  obtain  the  equivalent  equation 

6  X  +  9  =  4  X  +  13.  (2) 

For,  as  is  easily  shown,  2  is  a  root  of  each  equation,  and,  as  will 
be  seen  later,  2  is  the  only  root  of  either. 

This  example  illustrates  the  following  principle  : 

If  both  members  of  an  equation  are  multiplied  by  the  same 
known  expression,  not  denoting  zero,  the  derived  equation  will 
be  equivalent  to  the  given  one. 

That  is,  if  C  represents  any  known  expression,  not  denot- 
ing zero,  the  equations 


A  =  B 

and 

CA=CB 

have  the  same  roots. 

(1) 

(2) 


Proof     Any  root  of  (1)  makes  A  =  B.  §  102 

But,  when  A  =  B,         CA=  CB.  §  32,  (iv) 

Hence,  any  root  of  (1)  is  a  root  of  (2). 

Conversely,  any  root  of  (2)  makes  CA  =  CB.  §  102 

But,  when  CA  =  CB,      A  =  B,      since  C  ^  0.  §  32,  (v) 

Hence,  any  root  of  (2)  is  a  root  of  (1). 

Therefore,  equations  (1)  and  (2)  are  equivalent. 

Ex.  1.   Solve  the  equation  (5  x  -  12)  -4-  6  =  (x  -  3)  ^  3.  (1) 

Multiply  by  6,                               5  x  -  12  =  2  x  -  6.  (2) 

Transpose  terms,                          5  x  —  2  x  =  12  —  6.  (3) 

Unite  terms,                                           3  x  =  6.  (4) 

Multiply  by  1/3,                                      x  =  2.  (5) 


92  ELEMENTS  OF  ALGEBRA 

Proof  of  equivalency. 

Equation  (2)  has  the  same  roots  as  (1)  by  §108,  'identical  ex- 
pressions. ' 

Equation  (3)  has  the  same  roots  as  (2)  by  §  106,  '  addition. ' 
Equation  (4)  has  the  same  roots  as  (3)  by  §105,  'identical  ex- 
pressions.' 

Equation  (5)  has  the  same  roots  as  (4)  by  §  108,  'multiplication.' 
Hence  the  one  and  only  root  of  each  of  these  equations  is  2. 


Ex.  2.    Solve  the  equation  ^^^      ^      ^  =  \. 

(1) 

Multiply  by  12,                3  (x  +  1)  -  4  (x  -  1)  =  12. 

(2) 

Remove  (),                         3a;  +  3-4x4-4  =  12. 

(3) 

Transpose  terms,                                3ic  —  4a;  =  12  —  3  —  4. 

(4) 

Unite  terms,                                                  —  x  =  5. 

(5) 

Multiply  by  -  1,                                             x  =  —  6. 

(6) 

Proof  of  equivalency.  Equation  (2)  has  the  same  roots  as  (1)  by 
the  principle  of  '  multiplication '  (§  108) ;  (3)  as  (2)  by  '  identical  ex- 
pressions'  (§105);  (4)  as  (3)  by  'addition'  (§106);  (5)  as  (4)  by 
'identical  expressions'  (§105);  and  (6)  as  (5)  by  'multiplication' 
(§  108).    Hence  the  one  and  only  root  of  each  of  these  equations  is  —5. 

The  two  following  applications  of  the  foregoing  principle 
are  very  important : 

(i)  When,  to  clear  an  equation  of  fractional  coefficients, 
we  multiply  both  members  by  the  L.C.M.  of  their  known 
denominators,  the  derived  equation  has  the  same  roots  as 
the  given  one. 

(ii)  When  the  sign  before  each  term  of  an  equation  is 
changed  from  +  to  — ,  or  from  —  to  -f-  (that  is,  when  each 
member  is  multiplied  by  —  1)  the  derived  equation  has  the 
same  roots  as  the  given  one. 

109.   Roots  introduced  or  lost. 

If  we  multiply  both  members  of  the  equation 

3x-7  =  2x  +  2  (1) 

by  the  known  expression  0,  we  obtain  the  identity 

(3ic-7)  x0=(2x  +  2)  xO.  (2) 


INTEGRAL  LINEAR  EQUATIONS  93 

Observe  that  (1)  restricts  x  to  the  one  value  9,  while  (2)  does  not 
restrict  the  value  of  x  at  all. 

Again,  if  we  multiply  both  members  of  the  equation 

6x-l=4x  +  3  (3) 

by  the  unknown  expression  a;  —  5,  we  obtain  the  equation 

(6x-l)(x-5)  =  (4x  +  3)(x-5).  (4) 

Equation  (4)  has  the  two  roots  2  and  5,  while  (3)  has  only  the  one 
root  2.  Hence,  the  root  5  was  introduced  into  (4)  by  multiplying 
(3)  by  the  unknoion  expression  x  -^  5. 

The  two  examples  above  illustrate  why  the  multiplier  in  §  108  was 
limited  to  a  known  expression,  not  denoting  zero. 

If  we  divided  identity  (2)  by  0,  we  would  obtain  equation  (1). 

If  we  divided  equation  (4)  by  x  —  5,  we  would  obtain  equation  (3), 
and  one  root  would  be  lost  by  the  operation. 

This  illustrates  why  the  divisor  in  the  following  article  is  limited  to 
a  known  expression  not  denoting  zero. 

110.  Division.  If  both  members  of  an  equation  are  divided 
by  the  same  known  expression,  not  denoting  zero,  the  derived 
equation  will  be  equivalent  to  the  given  one. 

That  is,  if  0  represents  any  known  expression,  not  denot- 
ing zero,  the  equations 

A  =  B  (1) 

and  A^C  =  B-^G  (2) 

have  the  same  roots. 

Proof     Any  root  of  (1)  makes  A  =  B.  §  102 

But,  when  A  =  B,    A-^C=B-i-C.  §  32,  (v) 

Hence,  any  root  of  (1)  is  a  root  of  (2). 

Conversely,  any  root  of  (2)  makes  A-t-C=B^C.    §  102 

But,  when  A^C=B-^C,     A  =  B.  §  32,  (iv) 

Hence,  any  root  of  (2)  is  a  root  of  (1). 

Therefore,  equations  (1)  and  (2)  are  equivalent. 


94  ELEMENTS  OF  ALGEBBA 


Ex.  1.    Solve  the  equation  (x  -  1)  (x  -  2)  +  5  =  (x  +  1)2. 

(1) 

Eemove  (),                             x^-Sx  +  2-{-6  =  x^  +  2x  +  l. 

(2) 

Tra.nspose  terms,                               — Sx  —  2x  —  1— 2  —  5. 

(3) 

Unite  terms,                                                —  6x  =  -Q. 

(4) 

Divide  by  -  5,                                                  x  =  ^. 

(5) 

Proof  of  equivalency.  No  root  is  lost  or  introduced  by  any  one  of 
the  operations  performed  on  the  members  of  the  equations  from  (1) 
to  (5);  hence,  the  one  and  only  root  of  (1)  is  |. 

Ex.2.    Solve3(x-l)-{3x-(2-x)}  =  5.  (1) 

Remove  (  ),  Sx -S  -  Sx  +  2  -  x  =  5.  (2) 

Transpose,  3  x  -  3  x  -  x  =  5  +  3  -  2  (3) 

Unite  terms,  —x  =  6.  (4) 

Divide  by  - 1,  x=-6.  (5) 

Proof  of  equivalency.  No  root  is  lost  or  introduced  by  any  one 
of  the  operations  performed  on  the  members  of  the  equations  from 
(1)  to  (5);  hence,  the  one  and  only  root  of  (1)  is  —  6. 

Observe  the  difference  in  the  meanings  of  equal,  identical, 
and  equivalent.  Equal  applies  to  numbers,  identical  to  ex- 
pressions, and  equivalent  to  equations.  Two  numbers  are 
equal  or  unequal,  two  expressions  are  identical  or  not  iden- 
tical, and  two  equations  are  equivalent  or  not  equivalent, 
i.e.,  have  or  have  not  the  same  roots.  We  should  not  apply 
the  word  equivalent  to  numbers  or  expressions. 

111.  Erom  the  examples  given  above,  it  will  be  seen  that 
the  different  steps  in  the  process  of  solving  a  linear  equa- 
tion are  the  following : 

(i)    Clear  the  equation  of  fractions,  if  there  are  any. 

(ii)   Remove  parentheses,  if  there  are  any. 

(iii)  Transpose  the  unknown  terms  to  one  member  of  the 
equation,  and  the  known  terms  to  the  other  member. 


INTEGRAL  LINEAR  EQUATIONS  95 

(iv)  Unite  like  terms j  and  divide  both  members  by  the  co- 
efficient of  the  unknown. 

Note.  In  order  to  form  the  habit  of  clear  and  accurate  thinking, 
the  pupil  should  at  first  state  the  operation  by  which  each  equation  is 
derived  from  the  preceding  one,  and  note  whether  by  this  operation 
any  root  is  lost  or  introduced. 

But  as  he  advances  he  should  perform  the  simpler  steps  mentally, 
and  apply  two  or  more  principles  at  the  same  time. 

112.  A  numeral,  or  numerical,  equation  is  an  equation  in 
which,  all  the  known  numbers  are  denoted  by  numerals. 

Exercise  37. 
Solve  each  of  the  following  numeral  equations : 

1.  3a;-5  =  2a;4-l.  8.   x- (4:-2 x)  =  7  (x-1). 

2.  3x-\-^  =  x  +  S.  9.   5(4-3a;)  =  7(3-4a;). 

3.  4:X-4:  =  x-7.  10.   4(l-a;)  +  3(2  +  a;)  =  13. 

4.  7  a; -5  =  a; -23.  11.   S(x -2)  =  2(x  -  S). 

5.  8  x  +  42  =  5  X.  12.    2  x  -  (5  x -\- 5)  =  7. 

6.  5  a; -12  =  6  a.- -8.         13.    3(.c  +  1)  =- 5(a;  -  1). 

7.  7a;  +  19  =  5a;  +  7.         14.    7  (a;  -  18)  =  3  (a;  -  14). 
15.    2(a;-2)  +  3(a;-3)  +  4(.r-4)-20  =  0. 

16. .  2  (a;  -  1)  -  3  (a;  -  2)  +  4  (a;  -  3)  +  2  =  0. 

17.  5a;  +  6(.T-f  l)-7(a;  +  2)-8(a;4-3)  =  0. 

18.  2a;-[3-|4a;  +  (a;-l)|-5]  =  8. 

19.  (x  -  1)  (x  -  2)  =  (x  +  3)  (x  -  4). 

20.  3  a^  =  (a;  +  1)'  +  (x  +  2f  +  (a;  +  3)1 

21.  (x  -  2)  (x  -  5)  4-  (x  -  3)  (x  -  4)  =  2  (a;  -  4)  (a;  -  5). 

22.  5(x-\-iy-\-7{x  +  sy  =  i2(x-^2y. 

23.    (x  -  1)  {x  -  4)  =  2  a;  H-  (x  -  2)  (x  -  3). 


96  ELEMENTS   OF  ALGEBRA 

24.  x/5-x/4:  =  l.  27.    2  x/3  +  5  =  5  x/6  +  A. 

25.  {x-l)/2-\-{x-2)/3  =  3.    28.    x/2-\-2x/3  =  5x/6  +  7. 

26.  2x/3  +  4.  =  5-{-x/3.         29.    3  x/A  ~  5  =  7  x/S  -  6. 

30.  |-(2-a;)- |-(5ir  +  21)  =  a!  +  3. 

31.  J(^  +  l)-|-^(a;  +  2)+i(a^  +  4)  +  8=:0. 

32.  K^-5)-K^-4)-i(^-3)-(x'-2). 

33.  ^(a)H-i)_J(2a;-i)  +  li-0. 

34.  (3x  +  5)/8  -  (21  +  x)/2  =  5x-  15. 

35.  (a^  -  2)/3  -  (12  -  x)/2  =(5x-  36)/4  -  1. 

36.  (a.-  -f  8)/4  -  (5  a.-  +  2)/3  =  (14  -  a;)/2  -  2. 

37.  (x  -  15)/4  -  (7  -  2  a.')/21  -  3  a;/14  +  1/2. 

38.  5[4-(3a?-l)]=6(a^-ll)  +  49. 

39.  (x  -  2)/4  +  1/3  -lx-(2x-  l)/3]-  0. 

40.  3ip8-(a^/8+24)]  =  3i(2i  +  a./4). 

41.  5(3x-5)-17  =  8(3x-5)-2  (3  x  -  5). 

Combine  the  terms  involving  Sx  —  6. 

42.  2  (i»  +  1)  -  3  (ic  +  1)  +  9  (x  +  1)  +  18  =  7  (a;  +  1). 

43.  X  (x  -\- 2) -^  x  (x -\- 1)  =  (2  X  —  1)  (^x -{-  3). 

44.  x'-xll-x-2(3-x)']  =  x-^l. 

45.  3  (x  -  1)/16  -  5  (a^  -  4)/12  =  2  (i»  -  6)/5  +  5/48. 

46.  0.5  a;  +  3.75  ==  5.25  ^  —  1. 

To  clear  of  fractions  multiply  by  4. 

47.  2.25  X  -  0.125  =  3x-\-  3.75. 

48.  0.25  X  +  4  -  0.375  x  =  0.2  ic  -  9. 

49.  0.375  X  -  1.875  =  0.12  x  +  1.185. 

50.  O.W  a;  +  1.2  -  0.875  a;  +  0.375  =  0.0625  x. 


INTEGRAL   LINEAR  EQUATIONS  97 

113.  A  literal  equation  is  an  equation  in  which  one  or 
more  of  the  known  numbers  are  denoted  by  letters. 

E.g.,  ax-\-2x  +  i  =  0  and  ax  -\-  b  =  ex  are  literal  equations. 

Ex.    Solve  the  equation  (2  ~  5x)/a  =  (ex  +  7)/b.  (1) 

Multiply  by  ab,  (2  —  5x)b  =  (ex  -\-  7)  a.  (2) 

Remove  (  ),  2b  -  6bx  =  aex  +  7  a.  (3) 

Transpose,  6bx  —  acx  =  7  a  —  2b.  (4) 

Unite  terms,  —  (5  b  +  ac)  x  =  7  a  —  2  &.  (5) 

Divkle  by  -  (5  6  +  «c),  x  =  ?A=lI_«.  (6) 

56  +  ac 

Proof  of  equivaleney.  No  root  is  lost  or  introduced  by  any  one  of 
the  operations  performed  on  the  members  of  the  equations  from  (1) 
to  (6);  hence,  the  one  and  only  root  of  (1)  is  jj^iven  in  (6). 

114.  Any  linear  equation  in  one  unknown  has  one,  and  only 
one,  root. 

Proof.  By  transposing  and  combining  terms,  any  linear 
equation  can  be  reduced  to  an  equation  of  the  form 

ax  =  c,  (1) 

which,  by  §§  105,  106,  is  equivalent  to  the  given  equation. 

Divide  by  a,  x  =  c/a.  (2) 

Equation  (2)  is  equivalent  to  (1)  by  §  110 ;  hence,  c/a  is 
the  one  and  only  root  of  equation  (1),  or  of  the  given  linear 
equation. 

If  c  =  0  and  a^O,  the  root  c/a  is  zero  (§  98). 

If  c  ^  0,  then  the  smaller  a  is,  the  larger  arithmetically 
is  the  root  c/a. 

Observe  that,  if  b  is  an  arithmetic  number,  the  linear 
equation  x  —  b  =  0  has  the  arithmetic  root  b,  while  the 
equation  x-\-b  =  0  has  no  arithmetic  root. 


98  ELEMENTS   OF  ALGEBRA 

Exercise  38. 
Solve  each  of  the  following  literal  equations : 

1.  ax -i- b^  =  bx -{- a\  3.    (a -\-b)x  +  (b  —  a)x  =  b^ 

2.  x/a  +  x/b  =  c,  4.    2(x  —  a)-\-3(x—2a)  =  2  a. 

5.  (a-\-  b)x-\-  (a  —  b)x  =  al 

6.  (a  +  bx)  (b  4-  ax)  =  ab  (x^  —  1). 

7.  (a  —  x)  (a  -\-  x)  =  2  a^  -{-  2  ax  —  x^. 

8.  i(x-\-a-\-b)-{-i{x-{-a-b)  =  b. 

9.  i{a-\-x)-\-l(2a-\-x)  -\-i(3a  +  x)=3a. 

10.  ica  -=-  6  +  a?6  ^  a  =  a^  +  &^. 

11.  (a  +  &aj)  (&  +  ax)  =  ab(x^  —  1). 

12.  (a^  +  x)  (b^  -\-x)  =  (ab  +  xf. 

13.  (a;  4-  a  +  ?>)'  +  (a^  +  a  -  ^')'  =  2  a^. 

14.  {x  -a)(x-b)-\-  (a  +  &)'  =  {x -\- a)  (x -\-  b). 

15.  aic  (cc  +  a)  +  bx  (ic  4-  6)  =  (a  +  ^)  (a?  +  a)  (cc  +  b). 

16.  What  kind  of  a  number  is  the  root  of  a  numeral 
equation?     Of  a  literal  equation? 

See  §  5. 


CHAPTER   VIII 

PROBLEMS  SOLVED  BY  LINEAR  EQUATIONS  IN  ONE 
UNKNOWN 

115.  Having  learned  some  of  the  properties  of  linear  equa- 
tions in  one  unknown,  we  return  to  the  subject  of  solving 
problems  by  equations,  which  was  introduced  in  the  first 
chapter.     Reread  §  19. 

Prob.  1.  A  has  $80,  and  B  has  §  15.  How  much  must  A  give  to  B 
in  order  that  he  may  have  just  4  times  as  much  as  B  ? 

Let  X  =  the  number  of  dollars  that  A  must  give  to  B  ; 

then  80  —  x  =  the  number  of  dollars  that  A  will  have  left, 

and  16  +  X  =  the  number  of  dollars  that  B  will  have. 

But  A  will  then  have  4  times  as  much  as  B  ;  that  is, 

80-a;  =  4(15  +  ic).  (1) 

From  (1)  X  =  4. 

Hence  A  must  give  ^4  to  B. 

Prob.  2.  A  man  has  10  coins,  some  of  which  are  half-dollars,  and 
the  rest  dimes,  and  the  coins  altogether  are  worth  $  6.  How  many 
has  he  of  each  kind  ? 

Let  X  —  the  number  of  half-dollars ; 

then  16  —  a;  =  the  number  of  dimes. 

The  X  half-dollars  are  worth  ^  x  dollars, 

and  the  16  —  a;  dimes  are  worth  ^^  (16  —  x)  dollars. 

Now  the  coins  altogether  are  worth  $  6  ;  hence 

ix  +  T^(16-x)=6. 
From  (1)  05  =  11,  the  number  of  half-dollars. 

.'.  16  —  X  =   6,  the  number  of  dimes. 


100  ELEMENTS   OF  ALGEBRA 

Note.  It  should  be  remembered  that  any  letter  as  x  always  denotes 
a  number,  and  not  a  concrete  quantity.  Observe,  also,  that  in  any 
problem  all  concrete  quantities  of  the  same  kind  must  be  expressed  in 
terms  of  the  same  unit ;  for  example,  in  each  of  the  above  examples 
all  sums  of  money  were  expressed  in  terms  of  the  unit,  one  dollar. 

Prob.  3.  A  father  is  7  times  as  old  as  his  son,  and  in  5  years  he 
will  be  4  times  as  old  as  his  son.     How  old  is  each  ? 

Let  X  years  =  the  son's  age  ; 

then  7  x  years  =  the  father's  age. 

Hence  (x  +  5)  years  =  the  son's  age  after  5  years, 

and  (7  ic  +  5)  years  =  the  father's  age  after  5  years. 

.-.  7aj  +  5  =  4(x  +  5).  (1) 

From  (1)  x  =  6.     .:  7x  =  35. 

Hence  the  son  is  5  years  old,  and  the  father  35. 

Prob,  4.  Divide  60  into  two  parts,  so  that  three  times  the  greater 
may  exceed  100  by  as  much  as  8  times  the  less  falls  short  of  200. 

Let  X  =  the  greater  part ;  then  60  —  x  =  the  less. 

Three  times  the  greater  part  is  3  x,  and 

3  a;  —  100  =  the  excess  of  3  a;  over  100. 

Eight  times  the  less  part  =  8  (60  —  x)  and 

200  -  8  (60  -  a;)  =  the  defect  of  8  (60  -  x)  from  200. 

But  this  excess  and  defect  are  equal ;  that  is, 

3x- 100  =  200 -8(60- x).  (1) 

From  (1)  X  =  36,  the  greater  number. 

.*.  60  —  X  =  24,  the  less  number. 

Prob.  5.  A  could  do  a  piece  of  work  in  14  hours  which  B  could  do 
in  6  hours.  A  began  the  work,  but  after  a  time  B  took  his  place, 
and  the  whole  work  was  finished  in  10  hours  from  the  beginning. 
How  long  did  A  work  ? 

Let  X  =  the  number  of  hours  that  A  worked  ; 

theu  10  —  X  =  the  number  of  hours  that  B  worked. 


G^ 


A 


PROBLEMS  SOLVED  BY  LmEAE   EQUATIONS      101 

Since  A  could  do  the  whole  work  in  14  hours,  in  1  hour  he  would 
do  1/14  of  it;  hence 

^  x  =  the  part  of  the  work  done  by  A  in  x  hours. 

Since  B  could  do  the  whole  work  in  6  hours,  in  1  hour  he  would 
do  1  /6  of  it ;  hence 

1(10  —  x)  =  the  part  of  the  work  done  by  B  in  10  —  x  hours. 

But  A  and  B  together  did  the  whole  work  ;  hence 

t1jX  +  K10-^)=1.  (1) 

From  (1)  z  =  7.     .-.  10  -  x  =  3. 

Hence  A  worked  7  hours,  and  B  worked  3. 

Prob.  6.  Find  the  time  between  5  and  6  o'clock  at  which  the  hands 
of  a  watch  are  together. 

Suppose  that  the  hands  are  together  at  x  minutes  after  5  o'clock. 

At  6  o'clock  the  hour-hand  is  25  minute-spaces  ahead  of  the  minute- 
hand  ;  hence,  while  the  minute-hand  moves  through  x  minute-spaces, 
the  hour-hand  will  move  through  a;  —  25  such  spaces.  But  the  minute- 
hand  moves  12  times  as  fast  as  the  hour-hand ;  that  is,  in  any  given 
time  the  minute-hand  passes  over  12  times  as  many  minute-spaces  as 
the  hour-hand.     Hence 

a;  =  12  (a;  -  25).  (1) 

From  (1)  x  =  27^p 

Hence  the  hands  are  together  at  27y\  minutes  past  5  o'clock. 

Exercise  39. 

1.  Find  two  numbers  whose  sum  is  72,  and  whose  dif- 
ference is  8.  Ans.  40  and  32. 

2.  Divide  25  into  two  parts  whose  difference  is  5. 

3.  Divide  12  into  two  parts  whose  difference  is  16. 

Ans.  14  and  —  2. 

4.  The  difference  between  two  numbers  is  8;  if  2  be 
added  to  the  greater,  the  result  will  be  3  times  the  smaller. 
Find  the  numbers. 


102  ELEMENTS   OF  ALGEBRA 

5.  A  man  walks  12  miles,  then  travels  a  certain  dis- 
tance by  train,  and  then  twice  as  far  by  coach  as  by  train. 
If  the  whole  journey  is  78  miles,  how  far  does  he  travel  by 
train  ? 

6.  Find  two  numbers  whose  difference  is  12,  and  whose 
sum  is  equal  to  ^  their  difference. 

7.  Find  a  number  such  that  the  sum  of  its  sixth  and 
ninth  parts  will  be  equal  to  15. 

8.  Find  the  number  of  which  the  eighth,  sixth,  and 
fourth  parts  together  make  up  —  13.  A71S.  —  24. 

9.  Find  a  number  such  that  ^  of  it  shall  exceed  ^  of  it 
by  2.  Ans.  —  35. 

10.  Two  numbers  differ  by  28,  and  one  is  -|  of  the  other. 
Find  them. 

11.  A,  B,  and  C  have  a  certain  sum  of  money  between 
them.  A  has  ^  of  the  whole,  B  has  \  of  the  whole,  and  C 
has  $  50.     How  much  have  A  and  B  ? 

12.  A  and  B  together  have  $  75,  and  A  has  $  5  more 
than  B.     How  much  has  each? 

13.  A  has  $  5  more  than  B,  B  has  $  20  more  than  C,  and 
they  have  $  360  between  them.     How  much  has  each  ? 

14.  A  has  $  15  more  than  B,  B  has  $  5  less  than  C,  and 
they  have  ^  Q^  between  them.     Hoav  much  has  each  ? 

15.  A  has  $100,  and  B  has  |20.  How  much  must  A 
give  B  in  order  that  B  may  have  half  as  much  as  A? 

16.  The  sum  of  two  numbers  is  38,  and  one  of  them 
exceeds  twice  the  other  by  2.     What  are  the  numbers? 

17.  Find  a  number  which  when  multiplied  by  8  exceeds 
27  as  much  as  27  exceeds  the  original  number. 

18.  Find  two  numbers  of  which  the  sum  is  31,  and  which 
are  such  that  one  of  them  is  less  by  2  than  half  the  other. 


PROBLEMS   SOLVED  BY  LINEAR  EQUATIONS      103 

19.  Divide  100  into  two  parts  such  that  twice  one  part  is 
equal  to  3  times  the  other. 

20.  Four  times  the  difference  between  the  fourth  and 
fifth  parts  of  a  certain  number  exceeds  by  4  the  difference 
between  the  third  and  seventh  parts.     Find  the  number. 

21.  Fifty  times  the  difference  between  the  seventh  and 
eighth  parts  of  a  certain  number  exceeds  half  the  number 
by  44.     Find  the  number. 

22.  A  father  is  4  times  as  old  as  his  son;  in  24  years  he 
will  be  only  twice  as  old.     Find  their  ages. 

23.  A  is  25  years  older  than  B,  and  A's  age  is  as  much 
above  20  as  B's  is  below  85.     Find  their  ages. 

24.  A's  age  is  6  times  B's,  and  15  years  hence  A  will  be 
3  times  as  old  as  B.     Find  their  ages. 

25.  Find  a  number  such  that  if  5,  75,  and  35  are  added 
to  it,  the  product  of  the  first  and  third  sums  will  be  equal 
to  the  second  sum  multiplied  by  the  number. 

26.  The  difference  between  the  squares  of  two  consecu- 
tive whole  numbers  is  121.     Find  the  numbers. 

27.  Divide  $380  between  A,  B,  and  C,  so  that  B  will 
have  i$30  more  than  A,  and  C  will  have  $20  more  than  B. 

28.  The  sum  of  the  ages  of  A  and  B  is  30  years,  and  5 
years  hence  A  will  be  3  times  as  old  as  B.  Find  their  pres- 
ent ages. 

29.  The  length  of  a  room  exceeds  its  breadth  by  3  feet ; 
if  the  length  had  been  increased  by  3  feet,  and  the  breadth 
diminished  by  2  feet,  the  area  would  not  have  been  altered. 
Find  the  length  and  breadth  of  the  room. 

30.  The  length  of  a  room  exceeds  its  breadth  by  8  feet ; 
if  each  had  been  increased  by  2  feet,  the  area  would  have 
been  increased  by  60  square  feet.  Find  the  dimensions  of 
the  room. 


104  ELEMENTS   OF  ALGEBRA 

31.  The  width,  of  a  room  is  |  of  its  length.  If  the 
width  had  been  3  feet  more,  and  the  length  3  feet  less, 
the  room  would  have  been  square.  Find  the  dimensions 
of  the  room. 

32.  A,  B,  and  C  have  $  1285  between  them ;  A's  share  is 
greater  than  |  of  B's  by  ^25,  and  C's  is  j\  of  B's.  Find 
the  share  of  each. 

33.  If  silk  costs  6  times  as  much  as  linen,  and  I  spend 
$66  in  buying  40  yards  of  silk  and  24  yards  of  linen,  find 
the  cost  of  each  per  yard. 

34.  If  $600  be  divided  among  10  men,  20  women,  and 
40  children,  so  that  each  man  receives  $  15  more  than  each 
child,  and  each  woman  receives  as  much  as  two  children, 
find  what  each  receives. 

35.  Divide  $152  among  5  men,  7  women,  and  30  chil- 
dren, giving  to  each  man  $4  more  than  to  each  woman,  and 
to  each  woman  3  times  as  much  as  to  each  child. 

36.  A  sum  of  money  is  divided  between  three  persons, 
A,  B,  and  C,  so  that  A  and  B  have  $60  between  them, 
A  and  C  have  $  65,  and  B  and  C  have  $  75.  How  much 
has  each? 

37.  A  dealer  bought  four  horses  for  $1150;  the  second 
cost  him  $  60  more  than  the  first,  the  third  $  30  more  than 
the  second,  and  the  fourth  $  10  more  than  the  third.  How 
much  did  each  cost  ? 

38.  Two  coaches  start  at  the  same  time  from  York  and 
London,  a  distance  of  200  miles,  travelling  one  at  9J  miles 
an  hour,  the  other  at  9J  miles  an  hour.  In  how  many  hours 
after  starting  did  they  meet,  and  how  far  from  London  ? 

Ans.  10|  hours  ;  98J  miles  from  London. 

39.  A  man  leaves  ^  of  his  property  to  his  wife,  l  to  his 
son,  and  the  remainder,  which  is  $2500,  to  his  daughter. 
How  much  did  he  leave  to  his  wife  and  son  each  ? 

Let  X  =  the  number  of  dollars  which  he  left  in  all. 


PBOBLEMS   SOLVED  BY  LINEAR  EQUATIONS      105 

40.  A  man  divided  his  property  between  his  three  chil- 
dren so  that  the  eldest  received  twice  as  much  as  the  second, 
and  the  second  twice  as  much  as  the  youngest.  The  eldest 
received  ^3750  more  than  the  youngest.  How  much  did 
each  receive? 

41.  A  third  of  the  length  of  a  post  is  in  the  mud,  a 
fourth  is  in  the  water,  and  5  feet  is  above  the  water.  Find 
the  length  of  the  post. 

42.  A  flock  of  sheep  and  goats  together  number  84. 
There  are  3  goats  to  every  4  sheep.  How  many  are  there 
of  each  ? 

43.  Find  the  time  between  3  and  4  at  which  the  hands 
of  a  clock  are  together. 

44.  A  can  do  a  piece  of  work  in  30  days  which  B  can  do 
in  20  days.  A  begins  the  work,  but  after  a  time  B  takes 
his  place,  and  the  whole  work  is  finished  in  25  days  from 
the  beginning.     How  long  did  A  work  ? 

45.  A  can  do  a  piece  of  work  in  20  days  which  B  can  do 
in  30  days.  A  begins  work,  but  after  a  time  B  takes  his 
place  and  finishes  it.  B  worked  10  days  longer  than  A. 
How  long  did  A  work? 

46.  One  number  exceeds  another  by  3,  while  its  square 
exceeds  the  square  of  the  second  by  99.     Find  the  numbers. 

47.  Of  two  consecutive  numbers,  ^  of  the  greater  exceeds 
I  of  the  less  by  3.     Find  the  numbers. 

48.  A  garrison  of  1000  men  having  provisions  for  GO 
days  was  reinforced  after  10  days,  and  from  that  time  the 
provisions  lasted  only  20  days.  Find  the  number  in  the 
reinforcement. 

49.  In  a  mixture  of  spirits  and  water  half  of  the  whole 
plus  25  gallons  was  spirits;  and  a  third  of  the  whole  minus 
5  gallons  was  water.  How  many  gallons  were  ther%  of 
each  ? 


106  ELEMENTS   OF  ALGEBRA 

50.  At  3  o'clock,  A  starts  upon  a  journey  at  the  rate  of  4 
miles  an  hour,  and  after  15  minutes  B  starts  from  the  same 
place,  and  follows  A  at  the  rate  of  4J  miles  an  hour.  When 
does  B  overtake  A  ? 

51.  A  fish  was  caught  whose  tail  weighed  9  pounds;  his 
head  weighed  as  much  as  his  tail  and  half  his  body,  and  his 
body  weighed  as  much  as  his  head  and  tail.  What  did  the 
fish  weigh  ? 

52.  Find  a  number  such  that  if  |  of  it  be  subtracted 
from  20,  and  ^j  of  the  remainder  from  i  of  the  original 
number,  12  times  the  second  remainder  shall  be  half  the 
original  number. 

53.  A  cistern  can  be  filled  in  half  an  hour  by  a  pipe  A, 
and  emptied  in  20  minutes  by  another  pipe  B ;  after  A  has 
been  opened  20  minutes,  B  is  also  opened  for  12  minutes, 
then  A  is  closed,  and  B  remains  open  for  5  minutes  more, 
after  which  there  are  13  gallons  in  the  cistern.  What  was 
the  capacity  of  the  cistern  ? 

54.  A  father  was  24  years  old  when  his  eldest  son  was 
born ;  and  if  both  live  till  the  father  is  twice  as  old  as  he 
now  is,  the  son  will  then  be  8  times  as  old  as  now.  Find 
the  father's  present  age. 

55.  If  19  lbs.  of  gold  weigh  18  lbs.  in  water,  and  10  lbs. 
of  silver  weigh  9  lbs.  in  water,  find  the  quantities  of  gold 
and  silver  in  a  mass  of  gold  and  silver  weighing  106  lbs.  in 
air,  and  99  lbs.  in  water. 

56.  The  sum  of  $  1650  is  laid  out  in  two  investments,  by 
one  of  which  15  per  cent  is  gained,  and  by  the  other  8  per 
cent  is  lost;  and  the  amount  of  the  returns  is  $1725.  Find 
the  amount  of  each  investment. 

57.  How  many  children  are  there  in  a  family,  if  each  son 
has  as  many  brothers  as  sisters,  and  each  daughter  has  twice 
as  many  brothers  as  sisters  ? 


PROBLEMS  SOLVED   BY  LINEAR  EQUATIONS      107 

116.  Interost  formulas.  In  problems  of  interest,  the  quan- 
tities involved  are  the  principal,  interest,  rate,  time,  and 
amount. 

Let  p  =  the  number  of  dollars  in  the  principal ; 

r=  the  rate,  or  the  ratio  of  the  interest  per  annum 
to  the  principal ; 

t  =  the  number  of  years  in  the  time ; 

i  =  the  number  of  dollars  in  the  interest  for  t  years 
at  rate  r. 

a  =  the  number  of  dollars  in  the  amount,  or  the  sum 
of  the  principal  and  the  interest ; 

then  /=prt;  (1) 

and  a  =  p-\-  prf.  (2) 

Proof.  The  interest  on  $p  for  one  year  is  $pr',  hence  i, 
or  the  interest  on  $p  for  t  years,  is  $j^?*i;  whence  (1). 

But  a=2J-\-i;  whence  (2). 

If  any  three  of  the  four  numbers  /,  p,  r,  t,  or  a,  p,  r,  t  are 
given,  the  fourth  can  be  found  by  solving  equation  (1)  or  (2). 

Ex.  Find  the  principal  that  will  amount  to  $  1584  in  5  years  4 
months  at  G  per  cent. 

Here  a  =  1584,  <  =  5|,  r  =  0.06.     Substituting  in  (2),  we  have 

1584  =  p  4- P  (0.06)  (5|)  =  1.32  p. 

.-.  p  =  1584  -  1.32  =  1200. 

Hence  the  principal  is  $  1200. 

Exercise  40. 

1.  Solve  i  =prt  forp,  r,  and  t. 

2.  Solve  a  =  p  -{-jvt  forp,  r,  and  t. 

3.  Find  the  interest  on  $  4760  for  4  years  6  months  at 
5 1  per  cent. 


108  ELEMENTS   OF  ALGEBRA 

4.  Find  the  amount  of  $  3500  for  5  years  4  months  at  6 
per  cent. 

5.  Find  the  interest  on  $  7240  for  3  years  3  months  at 
8  per  cent. 

6.  The  interest  on  $  1250  for  8  months  is  $  50.     Find 
the  rate  per  cent. 

7.  The  amount  of  $  1050  for  2  years  6  months  is  $  1260. 
Find  the  rate. 

8.  The  interest  on  $3420  at  6  per  cent  is  $049.80. 
Find  the  time. 

9.  A   sum   of   money   doubles  in   12   years  6  months. 
Find  the  rate. 

10.  Find  the  principal  that  will  yield  f  262.50  per  month 
at  7  per  cent. 

11.  Find  the  time  in  which  $  1350  will  amount  to  $  1809 
at  6  per  cent. 

12.  The  interest  in  4  years  3  months  at  4  per  cent  is 
$  2099.50.     Find  the  principal. 

13.  Find  the  time  in  which  a  sum  of  money  will  double 
itself  at  6  per  cent. 

14.  The  interest  on  $  1270  for  8  months  is  $76.20.    Find 
the  rate. 

15.  At  4  per  cent  how  much  money  is  required  to  yield 
$  2500  interest  annually  ? 


CHAPTER   IX 
POWERS,  PRODUCTS,  QUOTIENTS 

117.  Certain  products  and  quotients  are  so  frequently 
required  in  Algebra  that  the  student  should  prove  and 
memorize  the  identities  by  which  they  can  be  written  out. 
In  this  chapter  the  most  important  of  these  identities  are 
proved,  and  used  in  obtaining  products  and  quotients. 

In  the  next  chapter  the  converses  of  these  identities  are 
used  for  factoring. 

118.  Power  of  a  power.  The  nth  power  of  the  mth  power 
of  any  base  is  equal  to  the  mnth  power  of  that  base;  and 
conversely. 

That  is,  (a/")"  =  a""". 

Ex.  1.       (23)2  =  23  X  23  =  23+8  _  26. 

Ex.  2.     (a2)*  =  a^a^a'^a^  =  a2+2+2+2  =  ^8. 

Ex.  3.     (a2)6  =  a-ixo  =  qjIj  .  conversely,  a^xc  =  {a^y. 

Proof     (a**)"  =  a'^a*"  •••  to  ?i  factors  by  notation 

—  ^m-f-m-f  •••ton  sunimandg  g    i^Ct 

=  a"•^ 

119.  Power  of  a  product.  Tlie  nth  power  of  the  product 
of  two  or  more  factors  is  equal  to  the  product  of  the  nth  powers 
of  those  factors  ;  and  conversely. 

That  is,  {aby  =  a"b",  (abc)"  =  a"b"c". 

Ex.1.     {ahcY  =  {abc){ahc){ahc)  by  notation 

=  {aaa)  (bbb)  (ccc)  by  laws  (A') ,  (B') 

=  a363c3,  by  notation 

109 


110  ELEMENTS   OF  ALGEBRA 

Proof. 
{aby'=(ah)  (ab)  •••  to  n  factors  by  notation 

=  (aa  •••  to  n  factors)  (bb  -■'  to  n  factors)     by  (A')^  (B') 
=  a^"".  by  notation 

Similarly  for  any  number  of  factors. 

Ex.  2.      (—  «)^  =  (—  1)^«^=  —  «^  ;  conversely,  (—  \ya^  =(— a)^. 
Ex.  3.     (2  x'-yy  =  2i(x-2)  V  =  16  x^y* ; 
conversely,  2*(x'^yy'^=  (2  x~yy. 

Ex.  4.        (-  5x'^y^y=  (-  5)3(x2)3(?/3)3^  _125iK6^9 

Exercise  41. 

Express  as  a  power  or  as  a  product  of  powers  each  of 
the  following  powers  of  products : 

1.  (-a^y.  7.  (aa^y.    '  13.  {ab-a^yy. 

2.  {-xy.  8.  {-Q^y.  14.  (-xh/'zy. 

3.  (-/)l  9.  (-aa;2)^  15.  (-2a5V)^ 

4.  (-z'f.  10.  {-bfy.  16.  (-2a26c«)^. 

5.  [(-a)^^  11.  (-2aary.  17.  {-^Ci'x^yy. 
6-  [(-2)^'.  12.  (-a2ic3)^  18.  {-a?Wxyy. 

19.  (-  a)^  (-  ay,  (-  a)^  (-  d)\  (-  a)«,  (-  a)^ 

20.  (-  aby,  (-  a5)3,  (-  aby,  (  -  a6)^  (-  aby. 

21.  (-2  a3z>4)2^  (_  2  a354^^3^  (_  2  a^by. 

22.  As  a  power  of  tlie  base  3^,  express  3^  3«,  S^^^  3^0,  S^-. 

23.  As  a  power  of  the  base  x^,  express  x'^,  x^,  x^,  x^^,  x^. 

24.  Asa  power  of  the  base  a?,  express  a^,  a}^,  a}^,  aP-,  oF*. 

Express  as  a  power  of  a  product : 

25.  6^x4^  27.    {-ay{-by       29.    {-xy%f{-zy 

26.  4^x(— 3)^       28.    (—ayb^  30.    a\x-\-yy. 


POWERS,    PRODUCTS,    QUOTIEXTS  111 

120.  Square  of  binomials.     By  multiplication,  we  obtain 

(a  +  6)-  =  fl2  +  2a6  +  6-.  (1) 

Tliat  is,  the  square  of  the  sum  of  tivo  numbers  is  equal  to 
the  sum  of  their  squares  plus  twice  their  product. 

Ex.  1.  (3  .X  4-  5  vY  =  (3  xy  +  (5  y^  +  2  (3  x)  (5  y)  by  (1) 

=  9  x2  +  25  y^  +  30  xy.  §  1 19 

Ex.2.  (2x-Syy=[{2x-)-{-(-Zy)Y 

=  (2x)2+(-3y)--^  +  2(2x)(-3y)  (2) 

=  4  a;2  +  9  2/2  _  12  xy.  (3) 

Ex.3.  (a_6)2^[«+(_5)-|2 

=  a2+(-^)2  +  2rt(-6) 

=  a2  -  2  ah  +  b^. 

In  the  examples  of  this  chapter  there  are  two  steps : 
First  step.     The  application  of  an  identity. 
Second  step.     The  simplification  of  the  result  obtained  by 
the  first  step. 

E.g.,  in  example  2,  the  application  of  identity  (1)  gives  the  result 
in  (2),  and  the  simplification  of  this  result  gives  (3). 

At  first  the  pupil  should  write  out  these  steps  separately  ;  later 
he  should  apply  the  identity  mentally,  and  write  only  the  final  result. 

Observe  the  advantage  gained  in  this  chapter  by  regarding  a  poly- 
nomial as  a  sum. 

121.  Square  of  polynomials.     If  in  the  identity 

{a  +  x)  -  =  a''^x'  +  2ax  (1) 

we  put  b  -{-y  for  x,  we  obtain 

(a  -h  6  -f /)-  =  a' ^(b-{-yy  +  2a(b+y) 

=  a-  +  b-  +/  +  2  a6  4-  2  a/  4-  2  6/.       (2) 
And  so  on  for  a  polynomial  of  any  number  of  terms. 


112  ELEMENTS   OF  ALGEBBA 

Hence  we  infer  that 

The  square  of  any  polynomial  is  equal  to  the  sum  of  the 
squares  of  its  several  terms,  plus  the  sum  of  the  products  of 
twice  each  term  into  each  of  the  terms  which  follow  it. 

Ex.    (x2  +  2?/-3c)2 

^(x2)2  4.(2?/)-^+(-3c)2  +  2a;2(2|/)  +  2x2(-3c)+2(22/)(-3c) 
=  X*  +  4  2/2  +  9  c2  +  4  ic^?/  -  6  cx2  -  12  cy. 

Exercise  42. 
Write  the  square  of  each  of  the  following  expressions : 

1.  2a  +  x.  18.    a«-2?/^  +  3c«. 

2.  X'  +  y-.  19.    ^x^  —  Qx  —  Q. 

3.  3a- 5  6.  'ZO.    x  +  y  +  z  +  v. 

4.  ^d^-Wx.  21.    x  +  y  —  z-v. 

5.  -2a2  +  56l  22.    x-y-z-v, 

6.  -aa^^  +  V-  23.    o.-^  +  a^^  -  2  cc  -  2. 

7.  3a6c-4a;V  24.    a  +  26-3c  +  4:d 

8.  -2  2;2-a&iKl  25.    1  +  a)  -  a^^  _|_  ^^ 

9.  -J^a?  +  ?>cz\  26.    3a2-a;2  +  c--2/. 

10.  a  +  h  +  c.  27.  4x2_3(^_4c_32/2. 

11.  a  +  &-c.  28.  3ir^-2a2  +  4  6-2A'. 

12.  a-6  +  c.  29.  8  o.-^/ -  4  a^^s. 

13.  a-h-c.  30.  |-a^2/^  +  f^y. 

14.  a +  26 +  4.  31.  2x-'»-7. 

15.  £c  +  2i/  +  3;2.  32.  ijx-y^-4.x^y\ 

16.  2  4-2a;-3a^.  33.  4  a'-^^  -  3  a'-'^^H 

17.  2a^-3a;-2.  34.  2(a  +  1)  -  5(6  +  c). 


POWERS,   PRODUCTS,   QUOTIENTS  113 

122.  Product  of  sum  and  difference.  By  multiplication,  we 
have 

(a+b)(a-b)  =  a'-b\  (1) 

That  is,  tJie  product  of  the  sum  and  the  difference  of  the 
same  two  numbers  is  equal  to  the  square  of  the  first  number  in 
the  difference  J  minus  the  square  of  the  second. 

Ex.    1.    (2x-^  +  5  6y3)(2x2-6  6y8)  =  (2a;2)2_(5  6?^)2  by  (1) 

=  4a:4_25&2y6  §§119,118 

By  properly  grouping  terms,  the  product  of  two  poly- 
nomials can  frequently  be  written  as  the  product  of  the  sum 
and  the  difference  of  the  same  two  numbers. 

Ex.  2.    (a  +  6  +  c){a  +  6  -  c)=[{a  +  h)+e]  [(a  +  6)-  c] 

=  (a  +  6)2-c2  by(l) 

=  a2  +  2a&  +  Z>2  -c2. 

Ex.  3.    (a  +  6-c)(a-6  +  c)  =  [a +  (6  -  c)J  [a -(6  -  c)] 

=  a2  -  (6  -  c)2 
=  a2  -  62  4.  2  6c  -  c2. 

Exercise  43. 
Write  each  of  the  following  indicated  products : 

1.  (b  +  a){a-b).  5.    (x'  +  4.if){x'-^f). 

2.  (5  +  a;)(a;-5).  6.    (3  or^  +  5  r')  (3  ar' -  5 /). 

3.  (l4-3ic)(l-3«).  7.    {^by-\-2ax)(2ax-Sby). 

4.  (62  +  a2)(a2_62).  8.    (4ca^  +  5  6-V)(4car-5i2^). 

9.    (a  4-  5  +  c)  (a  -  ft  -  c). 

10.  (l  +  6_c)(l-6  +  c). 

11.  (a-6  4-c)(a-ft-c). 

12.  (a;  +  32/-22;)(.^•-3?/-f-2^). 

13.  (or^  +  a^?/  +  /)  (x"  -xy  +  f). 

14.  (2r'  +  2/  +  2)(/-?/  +  2). 


114  ELEMENTS   OF  ALGEBRA 

15.  (3a  +  6-3c)(3a-6  +  3c). 

16.  (a2  +  3a-l)(a2-3a-l). 

17.  (a^-2a'-\-l)(a'-{-2a'-l). 

18.  (a^  -  62  _  c2)  (a2  -{- b^ -\-  c^). 

19.  (_a^_/  +  7)(a^-2/^  +  7). 

20.  (ax  —  hy-\-  cz)  {ax  +  hy  —  cz). 

21.  (3a;  +  9-42/)(3a;-9+42/). 

22.  (l4-4a;  +  32/  +  22)(l+4ic— 32/-20). 

23.  (a)  +  22/  +  «-6)(a;  +  22/-a  +  &). 

123.   By  multiplication,  we  obtain 

{x  +  a){x^b)  =  X'  +  (a  +  6)  jr  +  ab. 

That  is,  ^/ie  product  of  two  binomials  having  the  same  first 
term  is  equal  to  the  square  of  the  first  term,  plus  the  sum  of  the 
second  terms  into  the  first  term,  plus  the  product  of  the  second 
terms. 

Ex.  1.  (x  +  7)  (x  +  5)  =  aj'-^  +  (7  +  5)  X  +  7  X  5 

=  a;2  +  12  X  +  35. 
Ex.  2.  (a:  -  7)  (a;  -  5)  =  [X  +  ( -  7)]  [a:  +  ( -  5)] 

=  x2  +  (-7-5)x+(-7)(-5) 

=  x2  -  12  X  +  35. 
Ex.3.  (a;  +  7)(x-5)=x2  +  (7-5)a;  +  7(-5) 

=  a;2  +  2  a:  -  35. 
Ex.4.  (x-7)(a;  +  5)=a;2_2x-35. 

Exercise  44. 
Write  each  of  the  following  indicated  products : 

1.  (x4-8)(a;  +  5).  4.    (a;  -  4)  (aj  +  11). 

2.  (a;  -  3)  (a;  +  10).  5.    (a +  9)  (a -5). 

3.  (a;  +  7)(a;-9).  6.    (a-8)(a  +  4). 


POWERS,   PRODUCTS,    QUOTIENTS  115 

7.  (a -6)  (a +  13).  9.    (a  -  9  6)  (a  -  8  6). 

8.  (x-3a)(x  +  2d).  10.    (3x -2y){Sx -\- y). 
{Sx-2y)iSx  +  y)  =  (Sxy-^(-2ij  +  y)Sx-^(i-2y)y 

=  9x^-3xy-2y-2. 

11.  (a -5  6)  (a +  10  6).  17.  {xy  -  7  ab)  (xy  -  2  ab). 

12.  (x'-6)(a^  +  4:).  18.  (x-4:ab)(x-h5ab). 

13.  (a2  +  2a;)(a2-5ic).  19.  (xz  -  9  ab)  {xz -\- 11  ab). 

14.  (a;?/ -  9)  (a.-2/ +  6).  20.  (a"  +  c)  (a"  -  6). 

15.  (xy-6ab)(^xy-\-2ab).  21.  (a;''+i-3)(ic"+i-8). 

16.  (ab-5)(ab-\-7).  22.  (ic2«-i-6-)(a.-2"-i  +  c-). 

23.    (iB2  +  42/  +  42)(a:2_5y_52)^ 

Regarding  4(y  +  2)  and  -  5(?/  +  z)  as  the  second  terms,  we  have 
[x2  +  4(2/  +  5!)][x-i  -  5(?/  +  2)]  =x*  -(y  +  ^)x2  -  20(?/  +  z)'^. 

24.  (a;  +  2/  +  3)(aj  +  ?/-5).     26.    (x'-y  -  9)(a;-y +  8). 

25.  (a  +  6-7)(a  +  6-8).     27.    (a-4-2  6)(a+6-26). 

124.   Cubes  of  binomials.     By  multiplication  we  obtain 

(fl  +  bf  =  fl^  +  3  a-6  +  3  a6-  +  6^  (1) 

That  is,  ^/ie  CM6e  of  the  sum  of  two  numbers  is  the  cube  of 
the  first,  jjIus  three  times  the  square  of  the  first  into  the  second^ 
plus  three  times  the  first  into  the  square  of  the  second,  plus  the 
cube  of  the  second. 

Ex.  1. 

(2x  +  3?/)3  =  (2x)3+3(2x)2(3y)+3(2x)(3y)2  +  (3y)3    by  (1) 

=  8  x3  +  36  x^y  +  54  xy'^  +  27  y\ 
Ex.  2. 

(2  X  -  3  a)3  =  [(2  X)  +  ( -  3  a)]3 

=  (2  x)8  +  3(2  x)2(-  3  a)  +  3(2  x)(-  3  a)2  +  (-  3  a)* 

=  8  x3  -  36  x2a  +  64  xa^  -  27  a^. 
Ex.  3. 

(a  -  6)8=  a3  -  3  a26  +  3  ab'^  -  b\ 


116  ELEMENTS   OF  ALGEBRA 

Observe  that  when  the  second  term  has  a  negative  numeral  coeffi- 
cient, each  even  term  in  the  result  contains  an  odd  power  of  this 
coefficient,  and  therefore  has  a  negative  numeral  coefficient. 

The  same  is  true  in  §  120. 

125.  The  operation  of  raising  a  number  to  any  required 
jjower  is  called  involution. 


Exercise  45. 
Write  out  the  cube  of  each  of  the  following  expressions : 

1.  aj-f-l.                    6.    a  — 2  b.  11.  2aa^-\-7n^n. 

2.  2x-j-a.                 7.    ax  +  hy.  12.  i  a V  -  |  6 Y 

3.  a +  3  6.                 8.    2  a  — 3  6c.  13.  2  a;" +  5  2/". 

4.  i»  — 1.                    ^.    x^-\-^a.  14.  3.'c'"?/"  +  al 

5.  3ic  — a.               10.    xy  —  4:ab.  15.  x"b  —  3ay''+\ 

126.   Powers  of  sums.     By  multiplication  we  obtain 
(fl-f-6y  =  a^  +  4a«6+    6a'b'-{^   ^ab'  +  b\ 
{a  +  6)^  =  a^  +  5  a'b  +  10  a'b'  +  10  a'b'  +    5  ab' -h  b'. 
{a  +  6/  =  a«  +  6  a'b  +  15  a'b'  +  20  a'b'  +  15  a'b'  +  6  ab'  +  b\ 

The  expression  obtained  by  performing  the  indicated 
operation  in  (a-\-by  is  called  the  expansion  of  (a  +  by. 
Thus,  the  second  member  of  each  of  the  above  identities 
is  the  expansion  of  its  first  member. 

By  inspection  we  discover  in  each  of  these  expansions 
the  following  laws  of  exponents  and  coefficients: 

(i)  The  exponent  of  a  in  the  first  term  is  equal  to  the 
exponent  of  the  binomial,  and  it  decreases  by  1  from  term  to 
term. 

(ii)  The  exponent  of  b  in  the  second  term  is  1,  and  increa^ses 
by  1  from  term  to  term. 


POWERS,   PRODUCTS,    QUOTIENTS  117 

(iii)  The  coefficient  of  the  first  tenn  is  1,  and  that  of  the 
second  term  is  the  exponent  of  the  binomial. 

(iv)  If  in  any  term  the  coefficient  is  multiplied  by  a^s 
exponent,  and.  this  jwoduct  is  divided  by  6's  exponent  plus  1, 
the  result  is  the  coefficient  of  the  next  term. 

E.g.,  in  the  expansion  of  (a  +  6)^,  from  the  second  term  5  a*6,  by 
(iv),  we  obtain  5  x  4  -=-  2,  or  10,  which  is  the  coefficient  of  the  third 
term.  From  the  third  term  10  a^'fe'^  by  (iv),  we  obtain  10  x  3 -4- 3, 
or  10,  which  is  the  coefficient  of  the  fourth  term ;  and  so  on.  It  can 
be  proved  that  the  above  laws  hold  for  any  power  of  a  binomial. 

In  the  expansion  of  (a  +  6)*  there  are  5  terms,  each  of  the  fourth 
degree  in  a  and  6,  and  the  first  two  coefficients  are  repeated  in  inverse 
order  after  the  third  term.  In  the  expansion  of  (a  +  6)^  there  are  6 
terms,  each  of  the  fifth  degree  in  a  and  6,  and  the  first  three  coef- 
ficients are  repeated  in  inverse  order  after  the  third  term. 

Observe  that  in  each  of  the  above  expansions : 

The  sum  of  the  exponents  of  a  and  b  in  any  terra  is  equal 
to  the  exponent  of  tlie  binomial. 

The  number  of  terms  is  equal  to  the  exponent  of  the 
binomial  plus  1. 

The  coefficients  are  repeated  in  the  inverse  order  after 
passing  the  middle  terra  or  half  the  number  of  terms,  so 
that  the  coefficients  of  the  last  half  of  the  expansion  can 
be  written  out  from  the  first  half. 

Each  expansion  is  homogeneous  in  a  and  h. 

Ex.  1.    (2x+3  6)4 

=  (2  x)4+4(2  a;)8(3  6)  +6(2  x)2(3  6)2+4(2  a;)(3  6)8+ (3  6)* 

=  16  X*  +  96  x36  +  216  a;262  ^  216  x62  +  81  6*. 
Ex.  2.    {2x-ay 

=  (2  x)8  +  6(2  xy{-  a)  +  10(2  x)X-  a^  +  10(2  a;)2(-  a)3 
+  5(2x)(-a)*  +  (-a)6 

=  32  x6  -  80  x<a  +  80  xH^  -  40  x%8  +  10  xa*  -  a\ 
Ex.  3.    (a  -  6)6 

=  a6  -  6  a66  +  15  a*62  -  20  a^h^  +  15  a%^  -  6  a66  +  6«- 


118  ELEMENTS   OF  ALGEBBA 

Observe  that  when  the  second  term  in  the  binomial  has  a 
negative  numeral  coefficient,  each  even  term  in  the  expansion 
contains  an  odd  power  of  this  coefficient,  and  therefore  has 
a  negative  numeral  coefficient.  Thus  the  expansion  of 
(x  —  yy  differs  from  that  of  (x  +  yY  only  in  the  signs 
before  the  even  terms. 

Exercise  46. 

Expand  each  of  the  following  powers : 

1.  (2x  +  iy.  5.    (2  a -3)1  9.  {2m^-any. 

2.  (2-3  7/y.  6.    {a'-\-hcy,  10.  {cC'-2hy. 

3.  {x^  +  xyy.  7.    (2a;  +  3)^  11.  {ceb-x'^yf. 

4.  {2x  +  af.  8.    (3a-2&)«.  12.  {^a-c  +  x'f. 

Smce  any  polynomial  can  be  written  as  a  binomial,  the  laws  in  §  126 
can  be  used  to  expand  a  power  of  a  polynomial. 

E.g.^  we  can  write, 
(2  a  -  c  +  a:-)3=  [(2  a  -  c)  +  x'^Y 

=  (2  a  -  c)3  +  3(2  a  -  c)2(x2)  +  3(2  a  -  c)  (a;2)2^  (.^2)3 
=  8  a3  -  12  a2c  +  6  ac^  -  c^  +  12  a^x^  -  12  acx'^ 
+  3  c2x2  +  6  rtx4  -  3  cx^  +  x6. 

13.  (a^  +  .T4-l)3.  15.    (3.T2-5i»  +  l)'l 

14.  (x2_^._^2)^  16.    (2;c-3« -1-6)3. 

127.  Two  powers  are  said  to  be  like,  when  their  exponents 
are  equal ;  and  unlike,  when  their  exponents  are  unequal. 

E.g.^  a"^,  x^  are  like  powers;  a^,  a^,  a*  are  unlike  powers. 

128.  By  6  of  §  6,  like  powers  of  equal  numbers  are  equal. 
Hence,  like  powers  of  identical  expressions  are  identical. 

129.  By  division  we  obtain, 

(fl4  _  ^4^^  ^(a-b)  =  a'  +  d'b  -j-ab'  +  b'; 

(flS  _  ^5)  ^(a-b)  =  a'  +  w'b  +  a'b'  +  ab^  +  b\ 


POWERS,   PRODUCTS,    QUOTIENTS  119 

From  the  above  identities  we  infer  the  following  theorem : 

The  difference  of  the  like  x>owers  of  any  tivo  numbers,  as  a 
and  b,  is  exactly  divisible  by  the  difference  of  the  numbers, 
taken  in  the  same  order ;  the  laws  of  exponents  and  coeffi- 
cients in  the  quotients  being  as  follows : 

(i)  The  exponent  of  a  in  the  first  term  is  1  less  than  the 
exx>onent  of  a  in  the  dividend^  and  it  decreases  by  1  from  term 
to  term. 

(ii)  The  exponent  ofb  is  1  in  the  second  term  and  increases 
by  1  from  term  to  term. 

(iii)  The  numeral  coefficient  of  each  term  is  +1.  The 
number  of  terms  is  equal  to  the  exponent  of  a  in  the  dividend. 

Or  stated  in  symbols  the  theorem  is 

°"~f  =  a"  ^+  a"--b  +  a"-^b''  -f  ...  +  ab"-'-+  b"-\     (1) 
a  —  b 

where  n  is  any  positive  integer. 

Proof     Multiplying  the  second  member  of  (1)  by  the 
divisor  a  —  b,  we  obtain  the  dividend  a"  —  6". 
Hence  (1)  is  an  identity  (§  83). 

Ex    1    27  a'^-/>'^_(.^>  «)«-?>« 
3a -6         (3a)- 6 

=  (3a)2  -f  (3a)6  +  62  =  9rt2  +  3a?)  +  &2. 
Ex.  2. 

•^?-^^^^l^^^^=  (2  a)4  +  (2  a)8(3  a-)  +  (2  a)2(3  a;)2+ 2  a(3  x)8  +  (3  a-)4 
(2  a) -(3  a;) 

=  16  a<  +  24  aH  +  36  a%2  +  54  ax^  +  81  «*. 

Observing  that  each  term  in  the  dividend  is  the  fifth  power  of  the 
corresponding  term  in  the  divisor,  we  write  the  quotient  by  taking  the 
proper  powere  of  2  a  and  3  x. 

The  quotient  is  homogeneous  when  the  dividend  is  homogeneous. 


120  ELEMENTS   OF  ALGEBRA 

Exercise  47. 
Write  each  of  the  following  indicated  quotients : 
,     ^x^-4.f  ^     a^-16  ,,     8aV-3437>3 


12. 


3.    -^ 8.    ^-  13. 


^^    ^^^ ^^^  g^    .^^ .  ^^^ 


9a;^- 

-4/ 

30^^- 

-22/ 

81a* 

-16  6« 

9a2 

-46« 

xY- 

-a« 

xy- 

■  a 

64aV 

-8a^ 

4a- 

-2a; 

l--7292/^ 

a*- 

16 

a  — 

■2 

81  a^ 

'-1 

3a 

-1 

a;^- 

32 

iC  — 

•  2 

32  or' 

'-1 

2x 

-1 

243  < 

:i^-32 

h' 

1-92/  3a-26 


130.   By  division,  we  obtain : 

^^^    I  (a*  -  6*)  --  (a  +  5)  =  a^  -  a-6  +  ^^^  -  h^. 
{a^  4-  53)  ^  (f^  ^  5)  ^  ^2  _  «5  _^  52. 


2aV 

-Ih 

64a«63 

-S^f 

4a^6 

-2x'y 

ar^"+2  - 

4 

a;«+i  - 

2 

^+3_ 

y,. 

a,r.+  l_ 

2,.. 

^4n+4  _ 

if" 

^n+1  _ 

r 

{{a 


+  6^)  --  (a  -h  6)  =  a*  -  a^ft  +  a^ft^  _  ah^  +  &*• 


From  the  identities  in  (i)  and  (ii)  we  infer  the  two  fol- 
lowing theorems: 

(i)    Tlie  difference  of  the  like  even  powers  of  two  numbers 
is  exactly  divisible  by  the  sum  of  the  numbers. 

(ii)   The  sum  of  the   like  odd  powers  of  two  numbers  is 
exactly  divisible  by  the  sum  of  the  numbers. 

In  each  quotientj  the  laws  of  exponents  and  the  number  of 
terms  are  the  same  as  in  §  129. 

The  numeral  coefficient  of  any  odd  term  is  -f- 1,  and  that 
of  any  even  term  is  —  1. 


POWERS,   PRODUCTS,    QUOTIENTS  121 

Or  stated  in  symbols,  when  n  is  even,  the  theorem  is 

°"'^f  =  a"-'  -  a"-'b  +  a"-'b' +  ab"-'  -  b"-\    (1) 

a  +  6 

and,  when  n  is  odd,  the  theorem  is 

^"~^f  =  a"-^  -  a"-'b  +  a"-'b' ab"'  -h  b"-\    (2) 

Proof.  Multiplying  the  second  member  of  (1)  or  (2)  by 
the  divisor  a  -\-  b,  we  obtain  the  dividend  a"  —  b"  or  a*  +  6^ 
Hence  (1)  and  (2)  are  identities  (§  83). 

j,^    J     16x4-81?/^_(2a;)*-(3y)* 
2x  +  Sy  2x  +  Sy 

=  (2  a:)8  -  (2  xy(Z  y)  -\-  (2  x)  (3  y)2  -  (3  y)» 
=  8  x3  _  12  x-2y  -1-  18  a-y2  -  27  ys. 

^^    2.   i«!^ijt21«je!^=(2a62)2_(2a&2)(6ca;3)  +  (6cx8)a 
2  aft*  +  C  ca:* 

=  4  a'^b*  -  12  aft2cx8  +  36  c'^x\ 

Exercise  48. 

Write  each  of  the  following  indicated  quotients : 

J     1  -  a'b^  ^     x'  +  l  jg     a' +  32 

'    l-\-ab'  '    x  +  l'  '     a-\-2' 

2aa;  +  32^*  *    x-\-2'  '      a6  +  3 

Oa^^-'-lGy^  g     1±A^'.  15     729  + 8  6« 

3aj3  +  42/  "  *    l+2a  '     9  +  2b^ 

^    a*b^-a^f'  ^Q    a^y>  +  216y'        ^^     a'%'^-\-S2a^ 

ab^-\-x^f'  '       xy-\-6z    '  '      a-b^'-^2x 

a.y-16m«         J  J     (W-hSc^.  ^j     16xy-2o6a\ 

xif  +  27n^'  '    rt-V  +  2c'  *       2x^-\-4:d' 

6     ?Vzil.  12     tl±l.  18     ^'^''  +  ^y"- 

19.   Make  a  list  of  the  identities  proved  in  this  chapter. 


122  ELEMENTS   OF  ALGEBBA 

'  131.  The  remainder  theorem.  The  result  obtained  by  sub- 
stituting a  for  X  in  any  integral  expression  in  x  is  the  same 
as  the  remainder  arising  from  dividing  the  expression  by 
X  —  a. 

E.g.,  dividing  2  x^  —  x^  —  6  by  x  —  2  until  the  remainder  does  not 
contain  aj,  we  obtain  the  remainder  6,  and  6  is  what  2  x^  —  x^  —  6 
equals  when  x  —  2. 

Again,  dividing  x^  +  a^  by  x  —  a,  we  obtain  the  remainder  2  a^, 
and  2  d^  is  what  x^  +  a^  equals  when  x  =  a. 

Proof     Let  P  denote  any  integral  expression  in  x. 
Divide  Phj  x  —  a  until  the  remainder  does  not  contain  x. 
Let  Q  denote  the  quotient  and  R  the  remainder ;  then 

P=q{x~a)  +  R.  (1) 

Let  P]„  (read  ^ P  for  x  =  a^)  denote  the  value  of  P  when 
a  is  substituted  for  x. 

Put  a  for  X  in  (1);  then,  observing  that  Q']a{a—a)  is 
zero,  and  that  R  does  not  contain  x,  we  have 

Pla^R-  (2) 

132.  The  factor  theorem.  If  any  integral  expression  in  x 
becomes  zero  when  a  is  substituted  for  x,  the  expression  is 
exactly  divisible  by  x  —  a. 

Proof  From  P]„  =Ri\i%  131,  it  follows  that  if  P],  =  0, 
the  remainder  is  zero,  and  the  division  is  exact. 

Ex.  1.  The  expression  x^  —  a^  becomes  zero  when  a  is  put  for  x ; 
hence  x^  —  a^  is  exactly  divisible  by  x  —  a. 

Ex.  2.  The  expression  x'^  +  y^  becomes  zero  when  —  y  is  put  for  x  ; 
hence  x'^  +  y"^  is  exactly  divisible  \)j  x  —  {—  y),  or  x  +  y. 

Ex.  3.  The  expression  a"  —  &«  becomes  zero  when  &  is  put  for  a  ; 
hence  a"  —  &»*  is  exactly  divisible  by  a  —  5. 

Ex.  4.  When  n  is  odd,  a"  +  6"  becomes  zero  when  —  &  is  put  for 
a  ;  hence  a"  +  6"  is  exactly  divisible  by  a  —  (—  &),  or  a  -\-h. 


POWERS,   PRODUCTS,    QUOTIENTS  123 

Exercise  49. 

By  §  132,  prove  that  each  of  the  following  dividends  is 
exactly  divisible  by  the  corresponding  divisor : 

1.  (a^_a._6)-f-(a;-3).       3.    (a^ _  14 a;  -  8) -- (a; - 4). 

2.  (a^-2a;-15)-=-(a;-5).       4.    (s^  -  Sx"  +  4:)^{x-{-l). 

5.  (2i>:*-3i>^-4:X-\-5)-h(x-l). 

6.  (x*-2a'a^-\-ia^x-3a*)-i-(x-a). 

7.  (x^-3a^x'-7a^x-6a'')-ir{x-\-a). 

When  n  is  odd,  by  §§  131  and  132,  prove: 

8.  a;"  +  a"   is   exactly   divisible  by   x -{-  a,   but  not  by 
X—  a. 

9.  af  —  a"   is  exactly  divisible  by  x  —  a,  but  not  by 
x  +  a. 

When  n  is  even,  prove : 

10.  x""  —  a"  is  exactly  divisible  by  both  x-\-a  and  x  —  a. 

11.  a;** -fa"  is  not  exactly  divisible  by  either  a;  +  a  or 
X  —  a. 

133.  The  following  examples  illustrate  how  the  formulas 
in  §  129  or  §  130  often  aid  in  writing  out  the  partial  quo- 
tient and  the  remainder,  when  a  division  is  not  exact. 

Ex.  1.    Divide  a'^  -{-  b^  by  a  -  b. 

Adding  to  the  dividend  zero  in  the  form  —  b^  +  b^,  we  have 
a^  +  fe^-a*  -  &2  +  2  62_^  ^  ^  ^   2  62 


a  —  b  a  —  b 

Ex.  2.   Divide  a^  +  1  by  a  -  1. 
Adding  to  the  dividend  zero  in  the  form  —  1  +  1 ,  we  have 


a— la— 1  a— 1 

Ex.3.  ?i±i  =  5izii±2  =  ^_^.  +  ^_l+^_. 

x-^\        x-\-\  x  +  l 

Ex.4.  «i+3^a»-8  +  11^^3  +  2«  +  4+^J- 


0-2 


a^  +  4 

x-2 

a;2  +  5 

x^l 

0.-^  +  3 

x-\ 

y?  H-  cc" 

124  ELEMENTS  OF  ALGEBRA 

Exercise  50. 

Write  the  partial  quotients  and  the  remainders : 

1.   :._^.  5.    t^.  9.    t±L 

x  —  2  x-\-l 

2.  -  ■  --  6.  ^^!^1±/.  10.  •:?i_±_^. 

ax  —  y^  X  —  a 

x'^  -{■■  a'^  x^  —  a^ 

X  —  a  x-\-  a 

4.    :^^.  8.    ^^  +  ^.  12.    ^^±^. 

X  —  a  x-\-  a  x—2 

Simplify  each  of  the  following  expressions : 

13.  (l  +  a)2-(l  +  a)(l-a).       15.    (a  +  h)\a-h)\ 

14.  {\-xy+(l+x'){l-x').     16.  (22/-3a)2(2  2/+3ay 

17.  {x  —  a){x-^a){x^  +  a^){x'^-\-a'^). 

18.  {x'  -  a-)  (aj«  +  a«)  (a;^  +  a^)  (a;^  +  a^). 

19.  (a;2  -  X  +  1)  (x"  ^x  +  l){x'-x^  +  1). 

20.  (x-\-y--\-  z)  {x  +  z  —  y)  {y -\- z  —  x)  {x-{-y  —  z). 

Write  each  of  the  following  indicated  quotients : 

2j     a%^'-a^y^  ^^     g^"  -  64  a;^»+« 

ab'^  -x?f'  '      a"'  -  2  a;"+^  ' 

22  64ft^^-729a;^»  512^  _  729/"+^ 
2a2  +  3ar'     '                         '       b^-^-3y^'+' 

23  XV  +  128  2^        ^5m^l0n._32a;5n^«+5 
it'?/  +  2  *  a"*62«,  _  2  a;n^»i+l 

2^     a^V"+&^^'"  2g     a;^"  +  (a  +  &y'"+^, 


CHAPTER   X 
FACTORS   OF  INTEGRAL  LITERAL  EXPRESSIONS 

134.  The  problem  of  multiplication  is  '  given  two  or  more 
factors,  to  find  their  product.'  The  converse  problem, '  given 
a  product,  to  find  its  factors,'  is  the  problem  of  factoring. 
Reread  §§  33,  117. 

Certain  forms  of  products  which  frequently  occur  are 
called  type-forms,  as  a-  +  2  ab  +  6?  or  a-  —  b\ 

135.  Any  monomial  is  readily  resolved  into  its  factors. 

E.g.,  the  factors  of  5a;  (a  +  y)  are  6,  x,  and  a  -\- y. 

The  factors  of  xy  are  a;  and  y  or  —  a;  and  —  y  ;  but  we  usually  use 
the  factors  x  and  y  because  of  their  simpler  form,  unless  there  is  some 
special  reason  for  using  —  x  and  —  y. 

Again,  the  factors  of  x^  are  x  and  x  or  —  x  and  —  x  ;  that  is,  x^  is 
the  square  of  x  or  —  a:. 

136.  The  converse  of  the  distributive  law  is 

ax  +  bx  -\-  ex  -\-  "  =(a  -{-  b  -h  c  -h'--)x.  (1) 

Hence,  anjj  factor  ichich  is  common  to  all  the  terms  of  a 
polynomial  is  a  factor  of  the  j^olynomial. 

Ex.  1.   Factor  3  ax^  +  0  a^x  -  9  a^x^. 

Here  .3  ax  is  seen  tp  be  a  factor  of  each  term  ;  hence 

Zax^-\-6a^x-9 a^x^  =  a;(3 ax)  +  2  a (3 ax)  +  ( - 3 a*a;*) (3 ax) 

=  (x-\-2a-Sa-^x^)Sax. 

Hence  the  required  factors  are  3,  a,  x,  and  x  +  2a  —  S a^7?. 

In  identity  (1)  the  letters  x,  a,  6,  c,  •••  can  stand  for  any  binomial 
or  polynomial. 

125 


126  ELEMENTS   OF  ALGEBBA 

Ex.2.    ractora;(« -3  6)-2?/(a -35). 
The  binomial  a  —  3  &  is  a  factor  of  each  term  ;  hence 
x{a-Zh)-2 y{a  -^h)  =  {x-2y) {a  -  3  6). 

Ex.3.         y -x-2a{y  -  x)^\{y -x)-2a{]} -X) 

=  {\-2a){y-x). 

Ex.  4.   Eactor  {y  -  x){a^  ^h)-2(y-  x)(a^  -  b). 

The  expression  =  (a2  _]_  ft)  (^  _  a;)  _  2  (a^  _  ft)  (^  _  a;) 

=  [a2  +  ft_2(«2_ft)](y_a;) 
=  (Sb-a'^){y-x). 

Ex.  5.     a2(n  -  a;)  -  6-(.x  -  «)  =  a2(w  -x)+  b'\n  -  x) 

=  («2+  ft'2)(^  _  x). 

Exercise  51. 
Factor  each  of  the  following  expressions : 

1.  3  a; +  3.  13.    2  a'*^/"  +  6  «"+y +^ 

2.  a^  +  5  a;.  A71S.  2,  a%  ?/".  1  +  3  a^ 

3.  ab-\-  be.  14.    aa;"*+y+^  -f  bx'^+^y''+\ 

4.  4  a^  -  6  a^d.  ^l^^s.  3^"*+^  2/"^^  «  +  bxy. 

5.  2ax-\-3x\  15.    6  ?/'«+-- 3  2/^ 

6.  7a«-21a2?).  I6.    8x2"-4a;». 

7.  a^-5a^22/  +  20ar^/.  17.    7a;-+i-14a^. 

8.  5  aa^  -  10 a^x  _  5  a^a^l         18.    x(a +  1)  -  y(a +  1). 

9.  38a«6«-57a^6l  19.    y  (x  -  a)  -  x  +  a. 

10.  3a3&-6a2&2  +  9a263.         20.   y  (a?  -  a)  -  (a  -  a^). 

11.  15  a^^  -  10  a^c  +  5  aU        21.   4  (a^  +  1)^  -  6  (a^  +  1). 

12.  H a'x -  4: a'y- 12  a'b'.         22.    x(y -bf  -  c(b  -  y). 


FACTORS   OF  EXPEESSIONS  127 

137.   Trinomials  of  the  type-form  a-  +  2  a6  -f  b\ 

The  converse  of  the  identity  in  §  120  is 

a^  +  2  a6  +  6'  =  (a  +  by.  (1) 

That  is,  a  trinomial,  tivo  of  ivhose  terms  are  the  squares  of 
two  numbers  respectively,  and  the  remaining  term  is  twice  the 
product  of  these  numbers,  is  equal  to  the,  square  of  the  sum  of 
these  numbers. 

Ex.  1.    Factor  9  x2  +  24  x  +  16. 

9  x'^  is  the  square  of  8  a;,  16  is  the  square  of  4,  and 

24ic  =  2.3x.4. 

.-.  0a;2  +  24x  + 16  =  (3ic  +  4)2.  (1) 

Or,  9  x2  is  the  square  of  —  3  x,  16  is  the  square  of  —  4,  and 

24a;  =  2(-3x)(-4). 

.-.  9x2-f  24x+16  =  (-3x-4)2.  (2) 

The  factors  in  either  (1)  or  (2)  are  correct,  but  unless  there  is  some 
reason  to  the  contrary  we  usually  take  the  simpler  factors  given  in  (1). 

Ex.  2.    Factor  36  «»  +  6*  -  12  a^b^. 

36  a*  is  the  square  of  6  a-  or  —  6  «-,  and  b*  is  the  square  of  b^  or 

To  obtain  the  term  —  12  a^b^  we  must  take  either  6  a^  and  —  b^  or 
-  6  a^  and  6- ;  that  is, 

-  12  a'62  =  2  .  6  rt2(-  62)^  or  2(-  6  a2)ft2. 

.-.  36  a*  +  6*  -  12  a262  =  (6  rt2  _  ^2)2^  or  (  -  6  a2  +  62)2. 

Any  polynomial  which  is  to  be  factored  should  be  first 
examined  for  any  factors  common  to  all  its  terms. 

Ex.  3.    -  3  a^  4-  3  a^b^  -  lb  a'^b^  =  -  3  a3(a2  -  10  ax^  +  25  6«) 

=  -  3  «3(a  -  5  63)2. 
In  identity  (1),  a  and  6  can  denote  any  binomial  or  polynomial. 

Ex.  4.    (x-2y)2  +  2(x-2y)(3?/-2x)  +  (3y-2x)2 

=  [(X  -  2  2/)  +  (3  y  -  2  x)]2 -  (y  _  x)2. 


128  ELEMENTS   OF  ALGEBRA 

Exercise  52. 

Factor  each  of  the  following  expressions : 

1.  a2  +  6a  +  9.  13.  4.xY-x^-^y^. 

2.  ic2_|_i2ic  +  36.  14.  Sx--4.x'-4:. 

3.  a'-  +  25  +  10a;.  15.  xY -\- x\j -^  \ xf. 

4.  x^ -\- 121  -  22  X.  16.  4.a'x^-\-4:abxy-{-by. 

5.  a2  +  49-14a.  17.  9  a"  +  25  b'^  -  30  ab. 

6.  a2_|.25-10a.  18.  25  a'^a^  -  30  a^b-x  +  9  b\ 

7.  l-8a;  +  16ar^.  19.  25a^ +  25b' -  50a'b. 

8.  4a2  +  962_12a6.  20.  a^  +  25 ?>«  -  10 a5^ 

9.  9  a*  +  24  a^d^  +  16  6*.  21.  ^a' +  ^^b^  -  ^^a^b. 

10.  x^-^^y^-^xy.  22.  4  a.y  -  4  a;y  +  ar^. 

11.  5a^-10a-6  +  56l  23.*  (a -\- bf -\- 2 {a  +  b)  +  1. 

12.  a^  -  6  a^^  +  9  afel  24.  (2x-a)2-8(a-2a;)4-16. 

(2  X  -  rt)2  -  8(a  -  2  x)  +  16  =  (2  X  -  «)2  +  8(2  a;  -  a)  +  16. 
=  (2x-a  +  4)2. 

25.    (aj^4-2aj.'?/  +  /)«  +  (a;  +  ?/)5l 

(a;2  +  2  x?/  +  ?/2)a  +  (x  +  j/) &2  =  (x  +  ?/)2a  +  (x  +  ?/)&2. 
=  (ax  +  ay  +  &2)(a;  +  2/). 

26.  x\x  +  2)-\-2(x-{-2y  +  2x{x  +  2). 

27.  7?i^  4-  2  m?i + n^  —p  {711  +  n) .   29.    x""  -\-2  x'^y"'  +  2/^"*. 

28.  a{b-c)-Q)'-2bc+c').      30.    36 a;'^+2  _  43 ^«+i  _l_  Ig ^n 

138.  A  perfect  square  which  contains  only  two  different 
powers  of  some  one  letter  can  often  be  reduced  to  the  type^ 
form  o?  -{-2ab  -\-W  by  first  writing  the  polynomial  in  de- 
scending powers  of  that  letter. 


FACTORS  OF  EXPRESSIONS  129 

Ex.  1.    Factor  x^  -h  y^  +  z-  -{- 2xy  -  2xz  -  2yz. 

The  expression  contains  only  two  different  powers  of  x  ;  hence,  we 
arrange  the  expression  in  descending  powers  of  x,  as  follows  • 

The  expression  =  x^  +  2x(y-z)-\-(y^-\-z'^  —  2  yz) 
=  x^  +  2xiy-z)  +  (y-zy 
=  (X  +  y  -  zy. 

We  could  have  arranged  this  expression  in  descending  powers  of 
y  or  z. 

Ex.  2.   Factor  a*  +  4  6*  +  9  c*  +  4  a-b^  -  0  a-c^  -  12  b^d^. 

Arranging  the  expression  in  descending  powers  of  a,  we  have 

The  expression  =  a*  +  2a^(2  b'^  -  3  c2)  +  (4  6*  -f  9  c*  -  12  b^c^) 

=  a*-\-2  a2(2  b^  -  3  c^)  +  (2  62  -  3  c2)2 

=  (a2  +  2  62  _  3  c2)2. 

Ex.  3.   Factor 

a6  _  2  a^  +  3  a*  +  2  a\b  -  1)  +  a^Cl  -  2  6)  +  2  a6  +  62. 

The  expression  contains  only  two  different  powers  of  6  ;  hence,  we 
arrange  it  in  descending  powers  of  6,  as  follows  : 

62  +  2  6(a3  _  a2  +  a)  +  (a6  _  2  a5  4.  3  a*  -  2  a3  +  a2). 

This  expression  is  a  perfect  square,  if  its  last  term  is  the  square  of 
a'  -  a-  +  a.    By  §  121,  we  have 

(a«  -  a2  +  rt)2  =  a«  -  2  a^  +  3  a*  -  2  rt''  -f  d^. 
Hence  the  given  expression  is  identical  with 

62  +  2  6(a8  -  a2  +  o)  +  (a^  -  a'^  +  a)2, 
or  (6  +  rt3  _  ct2  4.  a)2. 

Exercise  53. 
Factor  each  of  the  following  expressions : 

1.  c^-6c{a-\-b)-\-9(a  +  by. 

2.  a*  +  6--|-4c2  +  2a6-f-'4«f  +  46c. 

3.  4a2  4-624-9c^  +  66c-12ac-4a6. 

4.  4  a*  +  6*  +  c-*  -  2  6-(r  -  4  aV  +  4  a-ft^. 


130  ELEMENTS   OF  ALGEBBA 

5.  a^ -\- 4:  y^  +  9  z^  +  4.  xy +  6  xz-^  12  yz. 

6.  25  a*  +  9  &^  +  4  c^  -  12  b'c'  +  20  c^a^  -  30  a'b'. 

7.  6  aca^  4-  4  5V  +  aV«  +  9  c-  -  12  ^ca^^  -  4  abx\ 

8 .  -  6  ft^c^  4-  9  c4  +  6^  _  12  c^a^  -f  4  a^  +  4  a-6l 

9.  6  aft^c  _  4  a^ftc  +  aW'  +  4  a^c^  +  9  b\^  -  12  abc\ 

Note.  The  products  in  exercise  53  can  be  factored  by  using  the 
converse  of  §  121. 

139.   Trinomials  of  the  type-form  jr-  4-  yojr  +  q. 
The  converse  of  the  identity  in  §  123  is 

X'  +  (a  -\-b)x-^ab  =  (x  +  a)  (x  +  b).  (1) 

Any  trinomial  in  the  form  x'  +^9a;  4-  g  can  be  written  in 
the  form  x^  -\-  (a  -{- b)  x  -{-  ab  and  factored  by  (1),  when  we 
know  the  two  factors  of  q  whose  sum  is  p. 

The  two  factors  of  q  whose  sum  is  p  can  often  be  found 
by  inspection  as  below  : 

Ex.  1.    Factor  x^  +  7  x  -{-  12. 

Here  p  =  7  and  q  =  12. 

The  two  factors  of  +  12  are  both  +,  or  both  —  ;  hence,  as  their 
sum  is  +  7,  both  are  +.  The  pairs  of  positive  whole  numbers  whose 
product  is  12,  are  12  and  1,  6  and  2,  4  and  3;  since  4  +  3  z=  7,  3  and  4 
are  the  two  factors  of  12  whose  sum  is  7. 

.-.  a;2  +  7  a;  +  12  =  x^  +  (3  +  4)a;  +  3  x  4 

=  (x+3)(x  +  4).  by  (1) 

Ex.  2.    Factor  x^-9x  +  20. 

The  two  factors  of  +  20  are  both  +  or  both  —  ;  hence,  as  their 
sum  is  —  9,  both  are  — .  The  pairs  of  negative  whole  numbers  whose 
product  is  20  are  —  20  and  —  1,  —  10  and  —2,-5  and  —  4  ;  since 
(—  5)+(—  4)  =  —  9,  — 5  and  —  4  are  the  two  factors  of  20  whose  sum 

is  -9. 

,,  a;2-9a;  +  20=  x2 +(- 5  -  4)a: +  (- 5)  -(-4) 

=  (a:-6)(x-4).  by  (1) 


FACTORS   OF  EXPRFSSIONS  131 

Ex.  3.    Factor  x^  +  6  x  -  27. 

The  two  factors  of  —  27  are  opposite  numbers  ;  hence,  as  their  sum 
is  +  6,  the  positive  factor  is  arithmetically  the  larger.  The  pairs  of 
whole  numbers  whose  product  is  —  27,  the  larger  arithmetically  being 
+ ,  are  27  and  —  1,9  and  —  3  ;  since  9  +  (  —  3)  =  6,  9  and  —  3  are  the 
required  factors. 

.-.  x2  +  6  a;  -  27  =  x2  +  (9  -  3)x  +  9 .  (  -  3) 

=  (x  +  9)(x-3).  by(l) 

Ex.  4.   Factor  a^x^  -  6  ax  -  84. 

The  two  factors  of  —  84  are  opposite  numbers  ;  hence,  as  their  sum 
is  —  5,  the  negative  factor  is  arithmetically  the  larger.  The  pairs  of 
whole  numbers  whose  product  is  —  84,  the  larger  arithmetically  being 
-,  are  -  84  and  +1,  -  42  and  +2,  -  28  and  +  3,-21  and  +  4, 
-  14  and  +  6,-12  and  7  ;  since  -  12  +  7  =  5,  -  12  and  +  7  are  the 
required  factors. 

.-.  (ax)2  -  5(ax)  -  84  =  (ax  -  12)  (ax  +  7). 

Ex.  5.   9  x*  -  12  X  -  77  =  (3  x)2  -  4(3  x)  -  77 
=  (3x-ll)(3x  +  7). 

Ex.  6.   Factor  x^  -  32  xy  -  105  y^. 

The  two  factors  of  —  105  y^  whose  sum  is  —  32  y  are  3  y  and  —  35  y. 
.-.  x2  -  32  xy  -  105  y'^={x  +  3  y)(x  -  35  y). 

Ex.7.  4a-a2  +  21=-(a2-4a-21) 
=  -(a-7)(a  +  3) 
=  (7-a)(a  +  3). 

Exercise  54. 
Factor  each  of  the  following  expressions  : 

1.  a^  4- 4  a; +  3.  6.  ar  +  2a;  — 3. 

2.  a^  — 4  a; -1-3.  7.  a:^-fa;  — 6. 

3.  a.-2-h9a;-|-20.  8.  a^-|-4a;-5. 

4.  a.'2_lla;-f-18.  9.  a^4-2a;-35. 

5.  x^-Sx  +  15.  10.  x'-Sx-lO. 


132  ELEMENTS   OF  ALGEBRA 

11.  x-x'-i-Q.  35.  4a?2_l2ic-91. 

12.  a^  +  5a;  +  14.  36.  oc^-20xy-96y\ 

13.  ar^  +  18i»4-72.  37.  x" - 26 xy  + 169 y^ 

14.  a;-ic2-M32.  38.  a^- 23  0^2/ +  132  2/'. 

15.  a^-5i»-84.  39.  4 a^  +  20 a??/ +  21 2/1 

16.  a^  +  5aj-150.  40.  9  of  -  39  xy -{- 22  y\ 

17.  a;2  _  25  a;  +  150.  41.  ar' +  43  a;2/ +  390  2/^^. 

18.  a^2  +  lla!-180.  42.  a" -20  abx-{- 75  b^x\ 

19.  a;-ar^-f-156.  43.  a^  -  29  a&  +  54  ftl 

20.  a;2_3ia;  +  240.  44.  130  +  31  a;2/ +  a^2/^. 

21.  a;2_34a;4.288.  45.  a^  + 12  a6aj  -  28  ?/-a.'l 

22.  a.-2-35a;-200.  46.  .'c^  + 13  aV  -  300  «^ 

23.  ar*- 17  a; -200.  47.  x^  -  a'x^  -  462  a\ 

24.  aV-21aa;  +  108.  48.  a;^  -  a^a^  -  132  a*. 

25.  aV  -  21  aa;  + 80.  49.  143  -  24  a;a  +  a^a^. 

26.  a^^  +  21  aa;  + 90.  50.  216 +  35  a? +  a^. 

27.  a^x^  - 19  aa;  +  78.  51.  e5  +  Sxy-  xy. 

28.  aV  +  30aa?  +  225.  52.  110  -  a? -a;^ 

29.  a^x^  +  54 aa;+ 729.  53.  98-7a?-a^. 

30.  aV  -  38  aa;  + 361.  54.  380-aj-a^. 

31.  a^y^-5xy-24:.  55.  120  -  7  aa;  -  a^a^. 

32.  4a:2^i2aj-55.  56.  105  +  16cy-cy. 

33.  9a^  +  6a;-35.  57.  (a; +  2/)^ +  6 (a; +  2/) +  8. 

34.  16a^  +  8a;-15.  58.  (a- 6)2+8 (a- 6) +15. 

140.  Trinomials  of  the  type-form  ax^  -\-  bx  -\-c. 

Multiplying  and  dividing  ax^  -{-bx  +  chj  a,  we  obtain 

aa^  +  &a;  +  c  =  \_{axy  +  h  (ax)  +  ac]  -4-  a.  (1) 


FACTORS   OF  EXPEESSIONS  133 

By  §  139,  the  trinomial  in  brackets  can  be  factored  by 
finding  the  two  factors  of  ac  whose  sum  is  b. 

Ex.  1.  3  ic2  _  16  X  +  5  =  [(3  a;)2  -  16(3  x)  +  15]  --  3 

=  (3a;- 16)(3x- l)-3  §139 

=  (x-5)(3x-l). 

Ex.  2.         6  x2  4-  32  a;  -  21  =  [(5  x)^  +  32(5  x)  -  105]  --  5 
=  (5  a;  +  35)  (5  X  -  3)  ^  5 
=  (a;  +  7)(5a;-3). 

Ex.  3.  3  x2  -  1*7  xy  +  10y^  =  [(3  x)^  -  17  y(3  x)+  30  2/2]  ^  3 
=  {x-6y)(Sx-2y). 

Exercise  55. 
Factor  each  of  the  following  expressions : 

1.  2x^-\-3x-{-l.  13.    3a^  +  13a;-30. 

2.  3a^  +  5a;-h2.  14.    Qa^-{.7x-S. 

3.  3a^  +  10a;  +  3.  15.    3 a^x^  +  23 ax -\- U. 

4.  3a^  +  8x  +  4.  16.    3 a^o.-^  +  19 oa;  -  14. 

5.  2ar^  +  7a;H-6.  17.   6  a'x'' -  31  ax -^  35. 

6.  2a^  +  llx  +  5.  18.    3a^  +  41a;  +  26. 

7.  6ar^-{- lla;  +  2.  19.   4a^  +  23a;  +  15. 

8.  2ic2_^3^_2.  20.    3ar_13a;  +  14. 

9.  4ic2  +  lla;-3.  21.    2a^-5xy-3f. 

10.  2xr-{-lox-S.  22.    3a^  -  17a;^  +  lOyl 

11.  3x^  +  7x-6.  23.    12ar'-23x?/  +  10/. 

12.  2ar^  +  a;-28.  24.    24 ar  _  29 a;?/ -  4 2/2. 

Factor  each  of  the  following  miscellaneous  expressions : 

25.  2a;(7i-l)-2(l-»).        28.    7 ar^  -  15 a?^  -  18 2/'. 

26.  c}j"'  —  ay'^+-+ny"'+\  29.    5x(a—2y)—2(2y—a). 

27.  9  «*  +  16  6^  -  24  a2^2^  30.    y?/^ -\-f/^^xy/3. 


134  ELEMENTS  OF  ALGEBRA 

31.  (a;-3)2  +  4(3-a:)  +  4.  37.  12^2  + 50aj  -  50. ' 

32.  1^2x^  +  x  —  l.  38.  aa^  +  (a  — 6)aj  — &. 

33.  aaj^  +  (a  +  6) a?  +  6.  39.  x^^2xy—4:XZ—4.yz+A:z\ 

34.  i»(a;  — a)2  — 2/(a;  — a).  40.  S?/"'-^— 32/*"+^+42/'"+^. 

35.  x^-^-^xr+^-5x^^\  41.  (a +  6)2 +  5  (a +  6) -24. 

36.  121  ar'  + 81/ +  198  a;?/.  42.  (a; -?/)'- 4  (a;  - 1/)  -  21. 

141.   Binomials  of  the  type-form  a"  —  6",  where  /i  is  even. 
The  converse  of  the  identity  in  §  122  is 

a^  -  6^  =  (a  +  b)  (a  -  b).  (1) 

That  is,  the  difference  of  the  squares  of  any  tivo  numbers 
is  equal  to  the  product  of  the  sumi  and  the  difference  of  the 
numbers. 

Ex.  1.   9  a%^  -  4  c2  =  (3  a%^y  -  (2  c)2 

=  (3  a353  +  2  c)  (3  a%^  -2  c)  by  (1) 

The  letters  a  and  h  in  (1)  stand  for  any  expressions. 

Ex.  2.    a2-4ay  +  4?/2-9c2=(a-2y)2_(3c)2 

=  (rt-2y  +  3c)(a-2?/-3c)  by  (1) 
Ex.  3.   9ic2+12a6-9a2-462  =  (3a;)2-(3a-26)2 

=  (3x  +  3a-2&)(3x-3a4-26). 

In  factoring  a  given  expression,  it  may  be  necessary  to  use  the  same 
principle  two  or  more  times  in  succession  as  below  : 

Ex.  4.      (X2  -  ?/  +  ^2)2  _  4  a;2^2 

=  (X2  -  ?/2  +  ^2  ^  2  XZ)  (X2  -  ?/2  +  ^2  _  2  OJ^) 

=  [(x  +  0)2_y2j[;(a;_^)2_y2j 

=  (x  +  ;s  +  y)  (x  +  ^  -  y)  (a:  -  ^  +  2/)  (x  -  0  -  ?/) . 

Whenever  n  is  even,  a""  —  6"  should  be  factored  as  the 
difference  of  two  squares. 
Ex.  5.   X*  -  «*  =  (a;2)2  _  (^2)2 

=  (X2  +  a-2)  (X2  -  ^2) 

=  (x2  +  a^)  {X  4-  a)  (x  -  a). 


FACTORS  OF  EXPRESSIONS  135 

Exercise  56. 
Factor  each  of  the  following  expressions : 

1.  a'-9,  6.   9a2-1662.        11.   Sab^-lSa\ 

2.  2oa'-b\  7.    81/-9a^.        12.    108  ar^  -  3  a.-^. 

3.  16-61  8.    36  ar- 49/.      13.    7a:^-28ar'. 

4.  a^- 9  2/2  9^    Aa-b''-9c\       14.    32  a^- 8  arY 

5.  64a.'2-49&l      10.   4a^-9a^.        15.    7xyz^-7a^f. 

16.  a^  +  2ab-{-b'--(^.  23.    49 ar' -  1  +  14 a;i/ +  ?/2^ 

17.  a2-2a64-&'-cl  24.    a^  _  ig  ^^.2  _|_  g  ^^  _^  9  ^2 

18.  a2_  j2_25c-c2.  25.    ar^- 9  ?/'+ 10  ax  +  25  a^. 

19.  a^-b^-{.2bc-c^.  26.    6"  -  a- -  4ar^  +  4oa;. 

20.  ar^  +  4a;i/-a=*  +  4/.        27.    9  0^- 4 a^- 9 a^^.  12 aa;. 

21.  ar^-l  +  10ca;  +  25c2.      28.    Actr  -  y- -9z^ -\-Qyz. 

22.  l-{-2ab-a'-b\  29.    c^  -  25a2_  952  ^30a6. 

30.  a^  +  b''-{-2ab-c^-(f-2cd. 

31.  a2  4-6--2a6-a^-/-2ajy. 

32.  m^-^ir-2mn-a^-b--{-2ab. 

33.  a2_,_,i2_2rt,i_^>2_^^2_2  5,,i. 

34.  16a2  +  8aa;  +  ar'-26v-62-2/l 

35.  9a'-\-12ab-\-4.b^-(c-{-x-2yy. 

36.  (a4-2'  +  c)2-ar-?/"  +  2a;?/. 

37.  (x  +  Syy-4:f.  39.    (5  a^  +  2 y)^  -  (3  a;  -  ?/)2. 

38.  9a2-(3a-56)2.  40.    (2a;+a-3)2-(3-2a;)2. 

41.  ixa*  —  ^\xb^    45.    5  — 80  a;*.  49.    a^"  — /". 

42.  |aV-fa/.     46.    aV  -  16  6y.     50.    9.T'*-a;"+2. 

43.  16  X*  —  y\  47.    .^•^"  —  /".  51.4  ar"+3  —  x'^+K 

44.  a* -81.  48.    ar''+2_y2n-2        53.    x^''+^y^-a^y\ 

53.  a;2  4-  /  +  ««  +  2  a^  —  2  a.-2;  +  2  2/2  —  16. 

54.  a*  +  4  62  +  9c2_  4a6  +  6ac- 126c -a:2_2a:2/_  2,2. 


136  ELEMENTS   OF  ALGEBRA 

142.  Binomials  of  the  type-form  a"  —  6",  where  n  is  odd. 
When  n  =  3,  by  §  129  we  have 

Ex.  1.   243-8  cfi  =  (7)3  -  (2  aY 

=  (7  -  2  a)  [72  +  7  (2  a)  +  (2  a)2] 
=  (7-2a)(49+14a  +  4a2). 
Ex.  2.    125  -  8  a^66  =  (5)3  _  (2  a262)8 

=  (5  -  2  a262)  [52  +  5(2  a262)  +  (2  a^lP')'^'] 
=  (5  -  2  a252)  (25  +  10  a262  +  4  a^M). 

Ex.3.    (1 -2a;)8-64x3  =  (l-2a;)3-(4x)3 

=  (l-2x-45c)[(l-2a;)2 

+  (l-2x)(4a:)  +  (4x)2] 
=  (l-6ic)(l  +  12a;2). 

When  n  =  5,  by  §  129  we  have 

a^-b'={a-  b)  {a'  +  a'b  +  a'b'  +  ab'  +  b'). 

Ex.  4.    2  a^  -  64  65  =  2  [a^  -  (2  6)5] 

^  2  (a  -  2  6)  [a*  +  a3(2  b)  +  a2(2  5)2 

+  a(2  6)8  +  (2  6)4] 
=2(a  -  2  6)  (a*  +  2  a35  +  4  ^252  +  8  aft^  +  16  6*). 

From  identity  (1)  in  §  129,  we  have 

a**  -  6"  =  (a  -  b)  (a«-i  +  a^'-^b  -\-a^-%^-\ h  ab""-^  +  &""0, 

when  n  is  any  positive  integer. 

143.  Binomials  of  the  t3rpe-form  a"  -f  6",  where  n  is  odd. 

When  n  ==  3,  by  §  130  we  have 

^3  +  53=  (aH-6)(a2-a6  +  62). 

Ex.1.    8a:3+27?/3  =  (2a;)3  +  (3?/)3. 

=  (2  a;  +  3  y)[(2  a:)2  -(2  x)(3  2/)  +  (3  ?/)2] 
=  (2  a;  +  3  ?/)  (4  a;2  -  6  a;?/ +  9  ?/2). 


FACTORS  OF  EXPRESSIONS  137 

When  n  =  5,  by  §  130  we  have 

a!^^b'=(a-\-  b)(a*  -  a^b  +  a'b^  -  ab^ -\-  6*). 
From  identity  (2)  in  §  130  we  have,  when  n  is  odd, 
a"  -f  6"  =  (a  +  b)  (a^^^  -  a'^-^b  H a6"-2  +  6"-!). 

Exercise  57. 
Factor  each  of  the  following  expressions : 

1.  a^-1.              9.   216-0)3.  17.  32a«  +  l. 

2.  27 -a^.           10.    27  718  +  1.  18.  a^&«  +  243. 

3.  a'-Sb^         11.   8a^-27a«.  19.  1024 ar"^  -  32 2/^. 

4.  125-a»6\      12.    a^6*'-ay8.  20.  x'-yl 

5.  x-^  +  l.             13.    40a3-13o6^  21.  x^ -1. 

6.  y3  +  27.           14.    27n3-f  64c».  22.  j;^ -f  128. 

7.  8ac»  +  64.         15.    2/^-1.  23.  1  -  (x  +  yf. 

8.  343- 8  al      16.   ar^- 32.  24.  x^  -  f. 

When  n  is  even,  x"  —  y»  should ^rs^  be  factored  as  the  difference  of 
two  squares  (§  141). 

a^  -  y«  =  (a;8  -  y3)(a;3  +  y3)  §  141 

=  (a;  -  y)(a;2  +  xy  +  y2)(x  +  y) (x*  -  xy  +  y^). 

25.  a;«-l.  28.    x^f-a^b\  31.    81aV-l. 

26.  a«-64.         29.    a;*-16  6\  32.    a«-729  6«. 

27.  »«-64/.      30.    16a;*-81a^        33.   64ic«-729/. 

34.  (3  +  2a)3-64.  38.  aV  -  (6  -  c)*. 

35.  a^-(x-{-yy.  39.  af^f-(xy  +  iy. 

36.  a;^_(a_6)«.  40.  16  a^  -  (?/ +  2  2)*. 

37.  a5«_^3._2  5y.  41.  27  aj^^  -  (a  +  &)«. 


138  ELEMENTS  OF  ALGEBRA 

144.   A  trinomial  of  the  type-form  a^  +  ha^b'^  +  6*  can  be 

factored  by  writing  it  as  the  difference  of  two  squares. 

Note.  The  two  factors  of  a*  +  ha'^lP'  +  6*  are  real  and  equal  when 
A  =  2  ;  real  and  unequal  when  ^  <  2,  and  complex  when  ^,  >  2. 

In  all  the  examples  given  the  factors  are  real  and  unequal,  but  as 
some  of  them  involve  surds  this  article  and  the  next  should  be  omitted 
until  Chapter  XVII.  has  been  studied. 

Ex.  1.   Factor  m*  +  m'^n'^  +  w*. 
Adding  mhi'^  —  mhi'^,  we  obtain 

wi*  +  n*  +  m^n^  =  m'^  +  w*  +  2  m^n"^  -  mH^ 
=  (m2  +  n^y  -  (mw)2 
=  (wi2  4-  n^  +  mn)  {m^  +  n^  —  m?i). 

Ex.  2.        m*  -  6  ni^n^  -\-  n'^  =  m*  +  #  -  2  m^n^  -Sm^ri^ 
=  (m2- w2)2  -(mny/S)^ 
=  (m'^-n^  + 17171  ^S){m^-n^-mn  V^) . 

Or,  m*  -  5  mhi^  +  w*  =  w*  +  w*  +  2  m2n2  -  7  m'^n^ 

=  (m2  +  n2  +  mw  V7)(to2+  ti^ -mn^Jl). 

Ex.  3.   4 x*  +  9 a*  -  21  a2ic2  =  4x*  +  9 a*  -  12  a2x2  -  9a2x2 
=  (2x2-3a2)2-(3ax)2 
EE  (2  a;2  -  3  a2  +  3  ax)(2  ic2-3  a2-3  ax). 

Exercise  58. 
"Factor  each  of  the  following  expressions  : 

1.  x^j^x'  +  l.  9.  25a3*-44i»2/  +  162/'. 

2.  .T*-3a^4-9.  10.  4x^-4a^/  +  92/^ 

3.  x*  +  9a^  +  25.  11.  ^x''-12x'y''  +  Uy\ 

4.  a;*  +  9fl^  +  25.  12.  16  x^  -  a^/ +  2/'- 

5.  a;*  -  11  aV  +  a^  13.  25  a;*  -  29  a^2/' +  4  .v'- 

6.  a;*  +  (4-c2)a^2/'  +  42/'.         14.  x^'-s^y^  +  y^. 

7.  (a;  +  2/y4-(a5  +  2/)'  +  l-         15.  aj^  +  a^?/' +  2/'- 

8.  9aj^  +  3a^?/2  +  42/^  16.  a^+a;y  +  /. 


FACTORS  OF  EXPBESSIONS  139 

145.  Binomials  of  the  type-form  a"  -f  6",  where  n  is  even. 

(i)      a*  +  b'  =  a*  +  2a-b^-{-b^-2d'b^ 
=  (a'  +  by  -  (ab^2y 
=  (a^  +  b'-\-  ab-y/2)  (a'  +  b'-  aby/2). 

This  method  can  be  employed  whenever  w  is  a  multiple  of 
4,  as  when  n  is  8,  12,  16,  etc. 

(ii)     a«  +  6«=(ay  +  W 

=  {a?  +  b^)[a^-o?b''+b''] 

=  {o?  +  b^lia'  +  by  -  (a6  V3)T 

=  (a2  +  b^  (a«  +  6^  4-  a6  V^)  («'  +  ?>'  -  a6  V3). 

This  method  can  be  employed  when  n  is  even  and  one 
of  its  two  factors  is  odd,  as  when  n  is  10,  12,  14,  etc. 

Exercise  59. 
Factor  each  of  the  following  expressions : 

1.  x^  +  1.                      5.    01? -a\  9.  3?-\-a\ 

2.  x^  H-  cy.                   6.    a^  +  1.  10.  a^  +  1. 

3.  nx^-\-a\                7.    a^  +  64.  11.  a;»«  +  ai«. 

4.  a^-1.                      8.    af-\-cy.  12.  a;"  +  a". 

146.  Perfect  cubes.    The  converse  of  identity  (1)  in  §  124  is 

a^  +  3a'b-\-3  ab^  +  b^  =  (a-\-  bf. 

Hence,  if  the  four  terms  of  the  cube  of  a  binomial  are  arranged 
according  to  the  powers  of  some  letter,  their  extreme  terms  are  the 
cubes  of  the  terms  of  the  binomial. 

E.g.,  if  64  a»  -  144  a'^b  +  108  ab^  -  27  68  is  a  perfect  cube,  it  is  the 
cube  of  4  a  —  3  6  ;  for  when  its  four  terms  are  arranged  in  descending 
powers  of  a,  the  extreme  terms  are  the  cubes  of  4  a  and  —  3  6 
respectively. 

The  expression  is  a  perfect  cube  ;  for 

(4  a  -  3  6)8  =  64  a^  -  144  a'^b  +  108  ab^  -  27  b^ 


140  ELEMENTS  OF  ALGEBRA 

If  a  perfect  cube  which  contains  only  three  different 
powers  of  some  letter  is  arranged  according  to  the  powers 
of  that  letter,  its  factors  will  often  become  obvious. 

E.g.,  if  we  arrange  the  expression, 

a^+b^  +  c^  +  S  a%  +  3  oT-c  +  3  a62  +  3  ac^  +  6  ahc  +  3  62c  +  3  ftc^, 
according  to  the  three  different  powers  of  a,  we  have 

a3  +  3  a2  (6  +  c)  +  3  a  (62  +  c2  +  2  ah)  +  (63  +  3  62c  +  3  6c2  +  c^), 
or  «3_^3rt2(54.c)  +  3a(6  +  c)2+(6+c)3, 

which  is  seen  to  be  (a  +  6  -f  c)^. 

Exercise  60. 

Factor  each  of  the  following  expressions : 

1.  a3  +  3a2-f-3a  +  l.         3.    Sm^  -  12m2 -f  6m  -  1. 

2.  a^  +  6a;2  +  12a;  +  8.        4.    aV -  3 a V^/^  +  3 aa;^^  - /. 

5 .  64  a«  +  108  ah^  -  144  a%  -  27  h\ 

6.  a;3- 24  0^2/ +  192  a?/ -512  2/3. 

7.    o?  +  6  a^ft  -  3  a^c  + 12  aV  -  12  a6c  +  3  ac'-{-  8  6^-12  6^0 

^_3^2     3^_^  8a^-4'ar^7/2+ 2a.y4_|!. 

842  J  -r^  y       21 

10.  24  62a.-3_36  6V  +  18  6=^a;-3  62. 

11.  a2_|_2a6  +  4c2  +  4ac  +  46c  +  6». 

12.  2  ax^  +  4  aa;22/2  -  4  aifz"  -\- 2  az^ -\- 2  ai/ -  4:  ax'z^ 

13.  3  6aj4  -  6  hxhj  +  12  a^^,  _  12  a^^/  +  3  62/'  -f  12  a6ar^. 

14.  a?x'-^d'x^y^-21a'xi/~21aY' 

15.  a^  +  3a^2/  +  3  V  -  3  aa.-'  J^f-^a^f-^  axy  +  3  a^2/ 
-{-Sa'x-a^ 


FACTORS   OF  EXPRESSIONS  141 

147.  Summary.  To  factor  any  given  expression  by  the 
foregoing  methods,  the  pupil  should  first  note  whether  the 
expression  is  in  any  one  of  the  following  forms  : 

(i)  A  sum  of  terms  having  a  common  factor.  §  136 

(ii)  A  perfect  power.  §§  137, 138, 146 

(iii)  A  difference  of  squares.  §  141 

(iv)  The  type-form 

ax^-\-bx-\-c  or  x^ -\- j^x -\- q.  §§  139,  140 

(v)  The  type-form  a"  -  ?/'  or  a"  +  6",  ?i  odd.     §§  142,  143 

(vi)  The  type-form 

a*  -f  /ia-62  +  h*  or  a"  +  6",  n  even.        §§  144,  145 

When  a  factorable  expression  has  no  one  of  these  forms, 
our  first  aim  is  to  reduce  it  to  one  of  them.  In  this  reduc- 
tion much  will  in  the  end  depend  upon  the  ingenuity  of 
the  student.  No  definite  directions  which  are  applicable  to 
all  cases  can  be  given.  The  two  following  devices  will  in 
many  cases  prove  useful : 

(i)  The  factors  of  an  expression  will  frequently  become 
obvious  when  the  exjJi'ession  is  aii'cmged  in  ascending  or 
descending  powers  of  one  of  its  letters^  particularly  when  the 
expression  contains  only  one  power  of  that  letter. 

Ex.  1.   Factor  ax  +  6y  +  6x  +  ay. 
Arranging  in  powers  of  x,  we  have 

ax -\- hy  -\- hx  +  ay  =  {a  -\- b)  x  ■{- {a  +  b)  y 
=  (a  +  b)(ix  +  y). 

Ex.  2.    Factor  ax^  —  x  —  a  -\-  I. 

Arranging  in  powers  of  a,  we  have 

ax3  -  X  -  a  -f  1  =  (x3  -  1)  a  -  (X  -  1) 

=  (x-l)[a(x2-Fx  +  l)-l] 
=  (x  -  l)(ax2  +  ax -I- a  -  1). 


142  ELEMENTS   OF  ALGEBRA 

Ex.  3.    Factor  «2  (^x-y)-\^  x^  (?/-«)+  y^  (a  -x). 

Arranging  in  powers  of  a,  we  have 

the  given  expression  =a'^(x  —  y)—  a (x^  —  y^)  +  xy  (x  —  y) 
=  (.^-y)  [«^  -(x  +  y)a-\-  xyl 
=  (x-y){a-x)(a-y). 

(ii)  Another  device  consists  in  adding  to  the  given,  expres- 
sion some  form  of  zero ;  as,  y^  —  y^,  or  —1+1. 

Ex.  1.   Factor  x'^  -  Sy^  -  z"^ -2xy  +  ^  yz. 

Arranging  in  descending  powers  of  x  and  adding  y^  _  y2^  y^^e  obtain 
the  given  expression  =x^  —  2  xy  +  y^  —  (4  y"^  -\-  z"^  -\-  i  yz) 
~{x-yy-{2y-zy 
=  (x  -  y  +  2  y  -  z)(x  -  y  -  2  y  +  z) 
=  (x  +  y  -  z)(x  -  S  y  i-  z). 

Ex.  2.    Factor  x^  -  3  ic  +  2. 

Adding  —  1  +  1,  we  obtain 

x^-Sx-\-2^(x^-l)-S(x-l) 

=  (x-l)(x^-]-x-]-l-S) 
=  ix-l)(x-l)(x  +  2). 

Ex.  3.  Factor  x^  -Sx^-{-4. 

Adding  x^  —  x^,  or  putting  —2x'^  —  x^  for  —  3  x"^,  we  obtain 
a;3  _  3  a;2  +  4  =  a;3  -  2  x2  -  x2  4-  4 

=  (X  -  2)  x2  -  (x2  -  4) 
=  (x-2)(x2-x-2) 
=  (x-2)(x-2)(x  +  l). 

Exercise  61. 
Factor  each  of  the  following  expressions : 

1.  a^  4-  a&  -h  oc  -h  be.  4.    mx  —  my  —  nx  -\-  ny. 

2.  orc^  4-  acd  +  ahc  -\-hd.       5.   Sax  —  bx  —  S  ay  -\-  by. 

3.  a^  H-  3  a  +  ac  4-  3  c.  6.   6oi^-{-Sxy  —  2ax  —  ay. 


FACTORS   OF  EXPRESSIONS  143 

7.  aaP  —  3  bxy  —  axy  +  3  by^. 

8.  2  ao/*^  +  3  axy  —  2  6x?/  —  3  by\ 

9.  ama:^  +  bma^  —  anxy  —  bny\ 

10.  ax  —  bx -{- by  -\-  cy  —  ex  —  ay. 

11.  a^x -\- abx -\- ac -\- dby -\- b^y -\- be. 

12.  iB3  +  ar^_4a;-4.  23.  2a;3  -  3ar^- 2a;  +  3. 

13.  57^-x^-^x-\-l.  24.  a:3  ^  5^  _  ^2^,  _  ^25^ 

14.  aa?-\-b3^-\-a  +  b.  25.  a^d^  _  a.2  -  52  4. 1. 

15.  aar^  +  by^  +  {a  +  b) xy.  26.  6a;^  +  ay?  +  bx-\-a. 

16.  a262  4_rt2_j_  j2^  1^  27.  x?  —  f-Yxz  —  yz. 

17.  a*  4- a'62  -  6 V  -  c^  28.  1  +  &^  -  (a' +  a6)ar'. 

18.  a''  —  a  —  c^  -}-  c.  29.  aV  +  acd  +  abc  +  bd. 

19.  a^  —  52  _  (a  —  6)2.  30.  ac-{-bd  —  ad  —  be. 

20.  a2-62  +  5c-ca.  31.  a<?  +  b(P  -  ad^  -  be\ 

21.  aa;^  +  a.-^  +  a  +  1.  32.  a^x  —  d^a;  +  a^?/  —  b^y. 

22.  ar'-Sar^  +  aj-S.  33.  a^ar' -  c'ar^  -  ay  4- c-y . 

34.  a'xf  -  ay  -  b'j^  +  by. 

35.  acar'  —  6ca;  -f  ada;  —  bd. 

36.  C'tf  -  C^  _  ^2g3^  ^  ^2^ 

37.  l-a6a;^  +  (&-a0^- 

38.  a^-b^-\-(r-d^-2(ac-bd). 

39.  4  a^ft^  _  (^2  _|.  ^2  _  ^2)2^ 

40.  (a'-b'^c'-cr'y-{2a^-2bdy. 

41.  a;^  +  a:'//  +  a^z'  +  2/2^.  43.    x*  —  14  a^y  +  y\ 

42.  a;(a;  +  2)-2/(2/  +  2)-  44.    xry^  -  x^z^  -  fz^  +  si^. 


144  ELEMENTS   OF  ALGEBRA 

45.  l  —  2ax  —  (c  —  a^) x^  +  aca?. 

46.  ax{f-\-h^)-\-hy{h^-\-a^y). 

47.  2ix?  —  4:  x^y  —  yrz  -\-2xy^  -\-2  xyz  —  yH. 

48.  {^^^xY-2{y?^^x)-lh. 

49.  (a2-2a)2-2(a2-2a)-3. 

50.  (i«2  4-4a;  +  8)2  +  3a?(a.'2-f  4a;  +  8)4-2ic2. 

51.  a^-6aj2  +  16. 

52.  a)3_15aj2  4-250.  53.    a^Y  +  13  icV  +  49 1/^ 

54.    36  a;Y  +  3  a.y  H- ?/^ 

55.  Resolve  c^  —  64  a^  —  a*'  4-  64  into  six  factors. 

56.  Resolve  ^  ■\-x^  —  16  ar^  —  16  into  five  factors. 

57.  Resolve  16  a;^  -  81  a;^  -  16  x""  +  81  into  five  factors. 

58.  Resolve  x^  +  aj^  +  64  x^  +  64  into  four  factors. 

59.  Resolve  x^  +  ^]f'  —  8  oi^if  —  Sy^  into  four  factors. 

60.  Factor  a~(b  -  c) -{- b^c  -  a) -{-  c'  (a-b). 

148.   Formation  of  equations  with  given  roots. 

The  linear  equation  whose  root  is  4  is  evidently  a;  —  4  =  0.         (1) 

The  linear  equation  whose  root  is  —  2  is  evidently  aj  +  2  =  0.      (2) 

Multiplying  together  the  corresponding  members  of  (1)  and  (2),  we 
obtain  the  quadratic  equation  (x  —  4)  (x  -\-  2)  =  0.  (3) 

When  x  =  4:,  (3)  becomes  the  identity  (4  -  4)  (4  +  2)  =  0. 

When  x  =  -2,  (3)  becomes  the  identity  (2  -  4) (-  2  +  2)  =  0. 

No  other  value  of  x  will  render  either  factor  in  (3)  equal  to  0. 

Hence  4  and  —  2  are  the  two  and  only  roots  of  (3). 

The  quadratic  equation  (3)  therefore  is  equivalent  to,  i.e.  has  the 
same  roots  as,  the  two  linear  equations  (1)  and  (2)  together. 
This  example  illustrates  the  following  principle : 


FACTORS   OF  EXPRESSIONS  145 

The  linear  equations 

x  —  a  =  0,    x—b  —  0,    X  —  c  =  0,  "'  (1) 

are  jointly  equivalent  to  the  equation 

(x  -a)(x-b){x-c)'"=  0.  (2) 

Proof.  The  root  of  any  one  of  the  equations  in  (1) 
renders  one  of  the  factors  in  (2)  zero;  hence  by  §  74  it 
satisfies  (2). 

Conversely  each  root  of  (2)  must  render  one  factor  of  its 
first  member  zero,  and  hence  be  a  root  of  one  of  the  equa- 
tions (1). 

Moreover,  equations  (1)  have  the  same  number  of  roots 
as  equation  (2). 

Hence  the  linear  equations  (1)  are  jointly  equivalent 
to  equation  (2). 

Ex.   Form  an  equation  whose  roots  are  1 ,  —  3,  and  4. 
The  linear  equations  whose  roots  are  1,-3,  and  4,  respectively  are 
X  -  1  =  0,     a;  +  3  =  0,    a;  -  4  =  0.  (1) 

By  §  148  the  equation  which  is  equivalent  to  equations  (1)  is 
(«-l)(a;  +  3)(x-4)  =  0, 
or  a;»-2x2-llx  + 12  =  0. 

Observe,  (i)  that  the  second  member  of  each  of  the  equa- 
tions (1)  and  (2)  is  0,  (ii)  that  equation  (2)  is  formed  from 
equations  (1)  by  multiplying  together  their  corresponding 
members,  and  (iii)  that  equations  (1)  are  formed  from  (2) 
by  putting  each  factor  of  its  first  member  equal  to  0. 

Exercise  62. 
Form  the  equation  whose  roots  are 

1.  -f-4,  +3.  5.    -2,  3. 

2.  -4,  +3.  6.    -2,-3. 

3.  2,  a  7.    -7,  4. 

4.  2,  -  3.  8.    1,  2,  3. 


9. 

1, 

-2, 

— 

3. 

10. 

— 

1,  - 

.2 

3. 

11. 

3, 

-4, 

5. 

12. 

1, 

-2, 

3, 

4. 

146  ELEMENTS   OF  ALGEBRA 

149.  To  solve  a  quadratic  or  higher  equation  we  must  find 
its  equivalent  linear  equations. 

For  use  in  solving  equations  the  principle  proved  in  §  148 
can  be  stated  as  follows : 

If  one  member  of  an  equation  is  zero  and  the  other  member 
is  the  product  of  two  or  more  integral  factors,  the  equations 
formed  by  putting  each  of  these  factors  equal  to  zero  are 
together  equivalent  to  the  given  equation. 

E.g.,  the  equation  (x  —  2)  (cc  +  3)  (x  —  4)  =  0 
is  equivalent  to  the  three  linear  equations, 

cc-2  =  0,  x  +  3  =  0,  a;-4  =  0. 

Ex.  1.    Solve  the  equation  x^  =  4  aj  +  12.  (1) 

Transpose,  x^  -  4  a;  -  12  =  0. 

Factor  the  first  member,  (x  +  2)  (x  -  6)  =  0.  (2) 

Equation  (2)  is  equivalent  to  the  two  linear  equations, 

x  +  2  =  0,  x-6  =  0. 
Hence,  the  roots  of  (2),  or  (1),  are  -  2  and  6. 

Ex.  2.    Solve  the  higher  equation  x^  +  x^  =  6  x.  (1) 

Transpose,  x^  +  x^  —  6  x  =  0. 

Factor,  x(x  -  2)  (x  +  3)  =  0.  (2) 

Equation  (2)  is  equivalent  to  the  three  linear  equations, 

x  =  0,  x-2  =  0,  x  +  3  =  0. 
Hence,  the  roots  of  (2),  or  (1),  are  0,  2,  and  —  3. 

Ex.  3.    Solve  the  equation  9  x^  =  4  x.  (1) 

Transpose,  9  x*  —  4  x  =  0. 

Factor,  x(3  x  +  2)  (3  x  -  2)  =  0.  (2) 

Equation  (2)  is  equivalent  to  the  three  linear  equations, 

x  =  0,  3x  +  2=0,  3x-2  =  0. 
Hence,  the  roots  of  (2),  or  (1),  are  0,  -  f,  and  f. 


FACTORS  OF  EXPRESSIONS  147 

Ex.  4.    Solve  the  equation  4  a.-*  +  9  =  37  x^.  (1) 

Transpose,  4  ac*  —  37  x^  _|_  9  -  o. 

Factor,         (2  x  -  6)(2  a;  +  6)(2  x  -  1)(2  x  +  1)=  0.  (2) 

Equation  (2)  is  equivalent  to  the  four  linear  equations, 

2x-G  =  0,  2x  +  6  =  0,  2x-l=:0,  2x+l=0. 
Hence  the  roots  of  (2)  or  (1)  are  3,  —  3,  I,  and  -  ^. 

These  examples  illustrate  the  following  rule  for  solving  a 
quadratic  or  higher  equation  in  one  unknown : 

Transpose  all  the  terms  to  one  member. 

Resolve  this  member  into  its  linear  factors  in  the  unknown. 

Solve  the  equations  formed  by  equating  to  zero  each  of  these 
linear  factors. 

The  problem  of  solving  an  equation  is  the  converse  to  that 
of  fanning  an  equation  with  given  roots. 

If  we  multiply  together  the  corresponding  members  of  equations 

X  -  3  =  1  and  X  +  3  =  16,  (1) 

we  obtain  x"2  -  9  =  16,  or  x^  -  25  =  0.  (2) 

The  roots  of  equations  (1)  are  4  and  13,  and  the  roots  of  (2)  are  6 
and  —  6. 

Hence  both  roots  of  equations  (1)  are  lost  by  multiplying  together 
their  corresponding  members. 

Putting  equations  (1)  in  the  form 

X  -  4  =  0  and  x  -  13  =  0, 

and  then  multiplying  them  together,  we  obtain  an  equation  equivalent 
to  equations  (1). 

This  illustrates  the  importance  of  the  form  of  the  equations  in  §  148 

Exercise  63. 
Solve  each  of  the  following  equations : 

1.  ar'-7if  =  0.  4.    x^-\-12x  =  -35. 

2.  x-  +  9x  =  0.  5.    x^  =  6x  +  91. 

3.  ic2  =  4a;-hl2.  6.   ar2  +  12  =  7a;. 


148  ELEMENTS   OF  ALGEBRA 

7.  a.'2  +  20  =  12a;.  33.  2x^-Sa^  =  5ax. 

8.  x'  +  20  =  9x.  34.  12x^  +  3a'  =  lSax. 

9.  a.'2  +  28  =  11  ic.  35.  132.^2  +  0^  =  1. 

10.  x^-\-150  =  25x.  36.  x^  +  G00a^  =  -^9ax. 

11.  3a;2  =  10a.'-3.  37.  or"^  -  3  a;^  =  10  a;. 

12.  3aj2_^lla^  =  20.  38.  16 .^'3 -f- 3 a;  =  16 «l 

13.  4a;2  +  21a^  =  18.  39.  110 aj^  +  a;  =  21  icl 

14.  3ar'-2a;  =  96.  40.  5a;3  =  8a^  +  21  a;. 

15.  15a^  +  4a'  =  3.  41.  32 a; - 3 a;»  =  10 a;*. 

16.  6ar^-7a;  =  3.  42.  x^ -\- 2  a^x  =  3  aa^. 

17.  19a;  =  4 -5a;2.  43.  a."^-a^H-9  =  9a;. 

18.  5x^-4.x  =  S3.  44.  .T3  +  2a^-16a;  =  32. 

19.  a;2  _^aa;  =  42  a^.  45.  x^-26x^-\-25  =  0. 

20.  a;2  _  20  aa;  =  96  a^  46.  aj*  +  36  =  13a^. 

21.  8a;2  +  a.'  =  30.  47.  36«*  + 1  =  13a;2. 

22.  a;  +  22  =  6a;l  48.  a;<- aV  +  4aV  =  4cV. 

23.  21-fa;  =  2ar^.  49.  a^  +  2a2  =  3aa;. 

24.  3x^  +  3o  =  22x.  50.  3562  =  9a^  +  66a;. 

25.  6  aj2^  55  a;  =  50.  51.  a^-2aa;  +  4a6  =  26a!. 

26.  6a;2_^6  =  13a;.  52.  3a^  -  2aa;  -  &a;  =  0. 

27.  19af'-39a;  =  -2.  53.  a^-2aa;  +  8a;  =  16a. 

28.  16x^-2ax  =  a\  54.  36 a;^ - 35 6^  =  12 6a;. 

29.  17a;2  +  8  =  70a;.  55.  x" -{-2(b -e)x-^c'  =  2bc. 

30.  21a:2^-|^()^^_;^  ^Q  x^-2(a-b)x-\-b^  =  2ab. 

31.  Gx'^llkx-^Tl^.  57.  (a-xy+(x-by=^(a-by, 

32.  a.-^  -  23  aa;  =  - 132  a^.         58.  a^ -]- x^  =  4:  x -\- 4:. 


FACTORS  OF  EXPRESSIONS  149 

69.  5x'^  —  x^  =  ox  —  l.  65.  bx^  +  ax^  =  bx -^ a. 

60.  x^  —  x  =  (^  —  c.  66.  x^—Sx^z=4:X  —  12. 

61.  x'-b'-^cx-bc.  67.  x* -\- 36  =  13 x". 

62.  2ic3_3^2^2a;-3.  68.  4x^  +  9=133^. 

63.  x^i-bx^  =  a^x  +  a'b.  69.  a;^  _^  o  a:^  ^  ;l^g  ^  ^  32, 

64.  x'-\-5  =  5x^  +  x.  70.  Oa:^  +  27.^^  =  .t- +  3. 

71.  Find  two  numbers  one  of  which  is  three  times  the 
other  and  whose  product  is  243. 

72.  Find  two  numbers  whose  sum  is  18  and  whose  product 

is  77. 

73.  A  certain  number  is  subtracted  from  36,  and  the  same 
number  is  also  subtracted  from  30 ;  and  the  product  of  the 
remainders  is  891.     Find  the  number. 

74.  A  rectangular  court  is  10  rods  longer  than  it  is 
broad;  its  area  is  375  square  rods.  Find  its  length  and 
breadth. 

75.  How  many  children  are  there  in  a  family,  when 
eleven  times  the  number  is  greater  by  five  than  twice  the 
square  of  the  number  ? 

76.  Eleven  times  the  number  of  yards  in  the  length  of  a 
rod  is  greater  by  five  than  twice  the  square  of  the  number 
of  yards.     How  long  is  the  rod  ? 

77.  The  square  of  the  number  of  dollars  a  man  possesses 
is  greater  by  1000  than  thirty  times  the  number.  How 
much  is  the  man  worth  ? 

Ans.  The  man  may  have  $  50  or  he  may  owe  f  20. 

78.  Find  two  numbers  the  sum  of  whose  squares  is  74, 
and  whose  sum  is  12. 


CHAPTER   XI 

HIGHEST   COMMON  FACTORS   AND  LOWEST  COMMON 
MULTIPLES 

150.  A  common  factor  of  two  or  more  expressions  is  an 
expression  which  will  exactly  divide  each  of  them. 

E.g.,  a  —  X  is  a  common  factor  of  6 (a  —  x)  and  a^  —  x^. 

151.  Two  or  more  expressions  are  said  to  be  prime  to  one 
another,  when  they  have  no  common  integral  factor  except  1. 

E.g.,  xy  and  vz,  3  aP'h  and  7  c^,  or  x^  +  y'^  and  x^  —  y"^,  are  prime  to 
each  other. 

152.  The  highest  common  factor  (H.  C.  F.)  of  two  or  more 
integral  literal  expressions  is  the  expression  of  highest 
degree  which  will  exactly  divide  each  of  them. 

The  numeral  factor  of  the  H.  C.  F.  is  the  greatest  common 
measure  (G.  C.  M.)  of  the  numeral  factors  of  the  given 
expressions. 

E.g.,  y?y'^z^  is  the  H.  C.  F.  of  x^^V  and  x'^y^z^. 

Again,  10  x^yz  is  the  H.  C.  F.  of  20  x'^yz  and  30  x^y'^z^. 

153.  H.  C.  F.  by  factoring. 

Ex.  1.   Find  the  H.  C.  F.  of  6  aP'hH^d^,  4  aHH,  and  8  a%&d^. 

The  H.  C.  F.  of  these  expressions  cannot  contain  a  higher  power 
of  a  than  a?-,  a  higher  power  of  c  than  <fi,  and  a  higher  power  of  d 
than  d  ;  and  the  G.  C.  M.  of  the  numeral  factors  is  2. 

Hence  the  H.  C.  F.  of  these  expressions  is  2  aP^c^d. 

Observe  that  the  power  of  each  base  in  the  H.  C.  F.  is 
the  lowest  power  to  which  it  occurs  in  any  of  the  given 
expressions. 

150 


HIGHEST  COMMON  FACTORS  151 

Ex.  2.    Find  the  H.  C.  F.  of  a*b-^  -  a^b*  and  a*b^  +  a^b*. 
a^b^  -  a%^  =  a%^{a  +  b)(a-b); 
a^b^-\-a^b*  =  a%^(a-\-  b). 
.'.  H.C.  F.  =  a-'62(rt  +  6). 

Ex.3.  Find  the  U.  C.F.olS a* +  \oa^b -12 a^b-2,  6 a^ -SO a'^b -\-S6  ab^ 
and  8  65  _  16  a*b  -  24  a^'^. 

3  a*  +  15  a^b  -  72  aHi^  =  S  -  a\a  4-  8  6)  (a  -  3  6) ; 

6  a3  -  3  a25  +  36a&2  =  6  •  a(«  -  2  6)(a  -  3  ft); 

8  a^  -  16  a45  _  24  a^P-  =  8  •  a3(a  +  &)  (a  -  3  6). 

.-.  H.C.  F.^rt(a-36). 

Hence,  to  obtain  the  H.  C.  F.  of  two  or  more  expressions, 
ivejind  the  product  of  their  common  factors^  each  to  the  lowest 
power  to  which  it  occurs  in  any  of  them. 

Exercise  64. 
Find  the  H.  C.  F.  of  each  of  the  following  expressions : 

1.  al)^,  a%.  10.  a;"?/"'-\  a;"-'?/"-^^  a;"+'?/"'. 

2.  a*6^  a%  ab\  11.  x" -{- jf\  x"  -  f. 

3.  a%x^,  ab'^x^,  a^b^x.  12.  a^  -  27,  9  -  al 

4.  3  a*,  2a^  4  a',  a\  13.  a*  -  ?/S  (a- -f /)'• 

5.  lOa;^  15ar*,  5.  14.  a'' —  b'^,  ax  —  bx. 

6.  lOd^y^,  20a;/e^  30ary.         15.  x'-l,x^-l. 

7.  3  3^2/2^,  l5  icyV,  10  a^/.  16.  a*^  +  8,  a'  -  a  -  6. 

8.  35  ay,  20  ay,  15  a^/a.         17.  a- -{- ab,  a" -{- b\ 

9.  a^y-,  a^^'»-\  a?y"+\  18.  a'-+3a;+2,  a^+6a;-|-8. 

19.  a."^  +  1,  a^  +  aar  +  aa;  +  1. 

20.  a;*  +  7  3^2  +  12,  a;*  +  6  .r^  +  8. 

21.  2&3 -h  3xhj  +  2  xy%  x'  +  6  ar^  +  8  3^y\ 


152  ELEMENTS   OF  ALGEBRA 

22.  3  a^  -  4  ab  -^b^Aa'-B  a^b  +  a'b''. 

23.  a^  —  a^x,  o?  —  ay?,  a*  —  aa^. 

24.  a?-l,x^-}-l,x^-2x-  3. 

25.  2  a^  -  7  X'  4-  3,  3  a;2  -  7  a.-  -  6,  4  a^  -  17  »  +  15. 

26.  12a;2  +  a;-l,  lo  x'' -\- S  x -^  1,  Ga^^  +  llaj  +  S. 

27.  2a;2  +  9a;  +  4,  2a;2  +  lla?4-5,  2a^-3a;-2. 

28.  a'x  -  a?bx  -  6  ab\  a'bx^  -  4  a6V  +  3  6'V. 

29.  a^  —  a^i/^  ^  +  ^V  +  ^2/  +  2/^?  ^^  —  2/^- 

154.  The  polynomial  factor  of  the  H.  C.  F.  of  two  expres- 
sions can  always  be  found  by  a  process  analogous  to  that 
employed  in  arithmetic  to  find  the  G.  C.  M.  of  two  numbers. 

This  process  depends  upon  the  two  following  principles : 

(i)  If  one  integral  expression  is  exactly  divisible  by  another, 
the  second  is  the  H.  C.  F.  of  the  two  expressions. 

E.g.,  (x^  —  y^)  -^{x^  +  xy  +  y'^)  =  x  —  y  ;  hence  by  definition  x^  + 
xy  -f  y'^  is  the  H.  C.  F.  of  x^  —  y^  and  x^  -+-  xy  +  y^. 

(ii)  If  one  integral  expression  is  divided  by  another  (of 
the  same  or  lower  degree  in  the  letter  of  arrangement),  and 
if  there  is  a  remainder,  the  H.  C.  F.  of  this  remainder  and  the 
divisor  is  the  H.  C.F.  of  the  first  tivo  expressions. 

E.g.,  the  remainder  obtained  by  dividing  the  expression 

a;3_2a;2_5x  +  6,  or  (x  -  l){x  +  2){x -S),  (1) 

by.  x'^-Sx-h  2,  or  (x  -  l)(x  -  2),  (2) 

is     .  _  4  a: +  4,  or  -4(a:-l).  (3) 

The  H.  C.  F.  of  the  remainder  (3)  and  the  divisor  (2)  is  evidently 
the  same  as  the  H.  C.  F.  of  the  two  expressions  (1)  and  (2). 

Proof  of  (ii).  Let  A  and  B  denote  any  two  integral  literal 
expressions  arranged  in  descending  powers  of  some  common 
letter,  the  degree  of  B  not  being  higher  than  that  of  A. 


HIGHEST  COMMON  FACTORS  153 

Let  Q  be  the  quotient  and  R  the  remainder  obtained  by 
dividing  Ahy  B\ 
then  A  =  BQ+R.  (1) 

From  (1)  R  =  A-  BQ.  (2) 

Every  factor  common  to  B  and  i2  is  by  §  136  a  factor  of 
BQ  -{-  R,  OT  A]  hence  every  factor  common  to  B  and  R  is 
common  to  A  and  B. 

Again,  every  factor  common  to  A  and  B  is  by  §  136  a 
factor  of  A  —  BQy  or  R;  hence  every  factor  common  to  A 
and  B  is  common  to  B  and  R. 

Hence,  the  H.  C.  F.  of  B  and  R  is  the  H.  C.  F.  of  A  and  B. 

The  following  example  will  illustrate  the  use  of  principles  (i)  and 
(ii)  in  finding  the  H.  C.  F.  of  two  expressions : 

Ex.  1.    Find  the  H.  C.  F.  of  x^  +  a;'  -  2  and  2i?  +  2x^-Z. 

Dividing  x^  +  2  a;^  —  3  by  x^  +  x^  —  2  we  obtain  the  remainder 
a;2  -  1. 

Hence,  by  (ii),  the  H.  C.  F  of  the  remainder  x^  —  \  and  the  divisor 
x*  +  x2  —  2  is  the  H.  C.  F.  of  the  two  given  expressions. 

Dividing  x^  +  x^  —  2  by  x^  —  1  we  obtain  the  second  remainder 
X  -  1.  Hence,  by  (ii),  the  H.  C.  F.  of  the  second  remainder  x  —  1,  and 
the  second  divisor  x^  —  1  is  the  H.  C.  F.  of  x*  +  x'*  —  2  and  x*  —  1, 
and  therefore  the  H.  C.  F.  of  the  two  given  expressions. 

But  x*  —  1  is  exactly  divisible  by  x  —  1  ;  hence,  by  (i),  x  —  1  is 
the  H.  C.  F.  of  x^  —  1  and  x  —  1,  and  therefore,  by  (ii),  of  the  two 
given  expressions. 

The  work  can  be  arranged  as  below  : 

a;8  +  x2  -  2)  x8  +  2  xa  -  3  (1 

X3  +   X2  -  2 

•   X2  -  1)  X8  +  X2  -  2  (X  +  1 
X*  -X 


x2  +  X  -  2 
X2     -1 


X  _  1)  x2  -  1  (X  +  1 


154  ELEMENTS   OF  ALGEBRA 

Before  employing  the  method  given  above,  all  monomial 
factors  should  be  removed  from  the  given  polynomials,  and 
the  H.  C.  F.  of  these  monomial  factors  found  by  factoring. 

Ex.  2.    Find  the  H.  C.  F.  of 

3  0%*  +  3  d^x^  -  6  c^x  and  6  cx^  +  12  ex*  -  18  cx^. 

3  c%4  +  3  c2x3  -  6  c%  =  3  c%  (x3  +  x2  -  2), 

and  6  c:*:^  _,.  12  ex*  -  18  cx^  =  6  cx^  (x^  +  2  x2  -  3). 

The  H.  C.  F.  of  the  monomial  factors  is  3  ex ;  and  by  example  1, 
the  H.  C.  F.  of  the  trinomial  factors  is  x  —  1. 

Hence  the  H.  C.  F.  of  the  given  expressions  is  3  cx(x  —  1). 

155.  The  H.  C.  F.  of  two  expressions  imll  not  he  changed  if 
either  expression  is  multiplied  or  divided  by  a  factor  which  is 
not  a  factor  of  the  other  exjjression. 

Proof  The  factor  introduced,  by  multiplication,  into  one 
expression  is  not  a  factor  of  the  other  expression,  and  there- 
fore will  not  be  a  factor  of  their  H.  C.  F. 

In  like  manner,  the  factor  removed,  by  division,  from  one 
expression  is  not  a  factor  of  the  other  expression,  and  there- 
fore would  not  be  a  factor  of  their  H.  C.  F. 

The  following  examples  illustrate  how  this  principle  frequently 
simplifies  the  work  of  finding  the  H.  C.  F.  of  two  expressions. 

Ex.  1.    Find  the  H.  C.  F.  of 

2  ax3  +  8  ax2  -  16  ax  +  48  a  and  4  ^2x4  _  4  a^x^  -f  32  a2x  -  32  a^. 
4  aH^  -  4  a^x^  +  32  a^x  -  32  a2  =  4  ^2  (^^a  __  ^rj  _|_  8  x  -  8), 

and  2  ax3  +  8  ax2  -  16  ax  +  48  a  =  2  a  (x3  +  4x2  -  8  X  +  24). 

The  H.  C.  F.  of  the  monomial  factors  is  2  a. 

To  find  the  H.  C.  F.  of  the  polynomial  factors  we  arrange  each 
expression  in  descending  powers  of  x  and  proceed  as  in  §  154. 

x3  +  4  x2  -  8  X  +  24)  X*  -  x3  +  8  X  -  8  (x  -  5 
xi  +  4x3-    8x24-24x 


-5x3+    8x2 -16  X-      8 
-  5  x3  -  20  x2  +  40  X  -  120 

28  x2  -  56  X  +  112  =  28  (x2  -  2  X  +  4) 


HIGHEST  COMMON  FACTORS  155 

By  §  155  we  reject  the  monomial  factor  28,  and  continue  the  process 
with  x2  —  2  X  +  4  as  tlie  second  divisor. 

x2  -  2  X  +  4)  x3  +  4  X--2  -  8  X  +  24  (x  +  6 
x8  -  2  x2  +  4  X 

6x2-12x  +  24 
6  x2  -  12  X  +  24 

By  (i),  x2  -  2  X  4-  4  is  the  H.  C.  F.  of  the  first  divisor  and  the  first 
remainder,  and  hence,  by  (ii),  of  the  polynomial  factors. 
Therefore  the  H.  C.  F.  sought  is  2  a  (x^  -  2  x  +  4). 

Ex,  2.    Find  the  H.  C.  F.  of 

2  x2  -  5  X  +  2  and  x^  +  4  x^  -  4  x  -  16. 

Since  2  is  not  a  factor  of  2  x^  —  5  x  +  2,  we  can  by  §  155  multiply 
x**  +  4  x2  —  4  X  —  16  by  2,  and  thus  avoid  the  inconvenience  of 
fractions. 

The  work  may  be  written  as  below  : 

2x2-5x  +  2)x8  +  4x2-  4x-16 
Multiply  by  2,        2 

2  x8  +  8  x2  _  8  X  -  32  (x 

2x8-5x2+  2x 

13x2-10x-32 
Multiply  by  2,  26  x2  -  20  x  -  64  ( 13 

26x2-65x  +  26 

Divide  by  46,  45)45x-90 

X  -  2)  2  x2  -  5  X  +  2  (2  X  -  1 
Hence  the  H.  C.  F.  is  x  —  2. 

Ex.  3.   Find  the  H.  C.  F.  of 

2x8  +  x2  -  X  -  2  and  3x3  -  2x2  -j-  x  -  2. 

Multiply  the  last  expression  by  2. 

2x8  +  x2  -  X  -  2)  6x8  -  4x2  +  2x  -  4  (3 
6x8  +  3x2-3x-6 

-7x2  +  5x  +  2 


156  ELEMENTS  OF  ALGEBBA 

Multiply  the  first  divisor  by  7. 

-7x2  +  5x  +  2)14a;3    +  7^2  -  7a;  -  14  (- 2x 


17x2_3x-14 
Multiply  this  remainder  by  7, 

119x2  -  21 X- 98  (-17 
119x2 -85x- 34 


Divide  by  64,  04)  64  x  -  64 


X-  l)-7x2+5x  +  2(-7x-2 

-  7  x2  +  7  X 

-2x4-2 
-2X-I-2 

Hence  the  H.  C.  F.  sought  is  x  —  1. 

In  the  above  process  of  finding  the  H.  C.  F.  of  two  inte- 
gral expressions,  each  remainder  is  evidently  of  a  lower 
degree  in  the  letter  of  arrangement  than  the  preceding  one. 
Hence  unless  at  some  stage  of  the  process  the  remainder 
is  zero,  we  must  come  at  last  to  a  remainder  which  does  not 
contain  the  letter  of  arrangement.  In  this  case  the  given 
expressions  have  no  common  polynomial  factor  containing 
that  letter;  for  by  §  154  this  last  remainder  contains  all 
the  polynomial  factors  common  to  the  given  expressions. 

156.  By  the  foregoing  principles  we  have  the  following 
rule  for  finding  the  H.  C.  F.  of  two  expressions : 

Remove  from  the  given  expressions  all  monomial  factors, 
and  set  aside  their  II.  C.  F.  as  a  factor  of  the  required  H.  C.  F. 

Divide  the  expression  of  the  higher  degree  arranged  in 
descending  powers  of  the  common  letter  of  arrangement  by  the 
other  expression ;  if  both  expressions  are  of  the  same  degree 
either  can  be  taken  as  the  first  divisor. 

Divide  the  first  divisor  by  the  first  remainder;  the  second 
divisor  by  the  second  remainder ;   and  so  on,  until  the  last 


HIGHEST  COMMON  FACTORS  167 

remainder  is  zero  or  does  not  contain  the  letter  of  arrange- 
ment. 

If  the  last  remainder  is  zerOj  the  last  divisor  is  the  H.  C.  F. 
sought;  if  the  last  remainder  is  not  zero,  the  two  expressions 
have  no  common  factor  in  the  letter  of  arrangement. 

Any  dividend  can  he  multiplied  by  any  number  which  is  not 
a  factor  of  the  corresponding  divisor;  or  any  divisor  can  be 
divided  by  any  number  which  is  not  a  factor  of  the  correspond- 
ing dividend. 

157.  Any  factor  common  to  three  or  more  expressions 
must  be  a  factor  of  the  H.  C.  F.  of  any  two  of  them. 

Hence,  to  find  the  H.  C.  F.  of  three  expressions,  we  can 
first  find  the  H.  C.  F.  of  any  two  of  them,  and  then  find  the 
H.  C.  F.  of  this  result  and  the  third. 

Ex.  Find  the  H.  C.  F.  of 

x»  +  ««  -  a;  -  1,  x»  +  Sa;*  -  X  -  3,  and  x«  +  iC''  -  2. 

The  H.  C.  F.  of  the  first  two  expressions  is  x^  —  1 . 
The  H.  C.  F.  of  x^  -  1  and  a:"  +  x-  -  2  is  a:  -  1. 
Hence  the  H.  C.  F.  sought  is  x  —  1. 

Whenever  the  given  expressions  can  be  factored  by  inspection,  their 
II.  C.  F.  should  always  be  obtained  by  factoring. 


Exercise  65. 
Find  the  H.  C.  F.  of  the  following  expressions : 

2.  x^  —  5xy  -^4:y%  x^  —  5x^y-\-4:  xif. 

3.  2x'-6x  +  2,  4.x^  +  12x'-x-^. 

4.  a^-5.T2-99.T  +  40,  .r»  -  6  a^  -  86  a; -h  35. 

5.  ^^2x'-Sx-lQ,  x^  +  ^o?-^x-24.. 

6.  ar'-a^-Saj-S,  a^-4iB2_iia;-6. 

7.  cBH3ar^-8a;-24,  a'»  +  3a^-3a;-9. 


158  ELEMENTS  OF  ALGEBRA 

8.  a^-{-3x'y-10afy%  x'-3x''y  +  2y^. 

9.  2a2-5a  +  2,  2a3-3a2-8a+.12. 

10.  262_5^,^_2,  12b^-Sb^-Sb  +  2. 

11.  a^-Ba^x-^Tax^-Sx^,  a^  -  3  ax" -\- 2  a^, 

12.  x'^-2x^-4.x-7,  x'^  +  x^-3x^-x  +  2. 

13.  r'-3a2i»-2a3^  .^-aa^--4a3. 

14.  2x'  +  4:a^-7x-U,  6a^-10x^-21x  +  35. 

15.  2a;^-2a.'3  +  aj2  +  3a;-6,  4a;4-2a^+3a?-9. 

16.  3aj3  +  a;2+x-2,  2a^-a^-a;-3. 

17.  3a^-3ax'  +  2a^x-2a^,  3 a^ -i- 12 aa^ -^  2 a^x -\- S a\ 

18.  3  a.*^  —  3  ay^y  +  a;?/^  —  2/^  4  a^^/  —  5  .ti/^  +  2/^. 

19.  12x^-15xy-{-3y'',  6a^-6a^y-^2xy^-2f. 

20.  10a^  +  25aa^-5a^  4a^4-9aa^-2a2aj-a3. 

21.  6a3+13a2a;-9aa.'2-10a;«,   9a3H-12a2.r-ll  aa^-lOa^. 

22.  2a;* 4-9x3+140^  +  3,  2  +  9a;  +  14ar^4-3a;^ 

23.  3a;*  +  5a:3_7^_^2a;  +  2,  2a;4  + 3a^- 2a;2  +  12a;  +  5. 

24.  2a^-lla^  +  lla;  +  4,  2aj*-3a^  +  7ar^  -  12a?-4. 

25.  2a;*  +  4a^  +  3a;2_2a._2,  3a;*  +  6ar^  +  7a;^  +  2aj  +  2. 

26.  x^  -  9  a;  -  10,  a^  -  7  X  -  30,  a^  -  11  a;  +  10. 

27.  x2_|_^_6^  ^_2a^-a?  +  2,  x«  +  3a^-6x-8. 

28.  ar^+7x-+5a;-l,  x2+3x-3x^-l,  3a^+5x-+a;-l. 

LOWEST  COMMON  MULTIPLE. 

158.  A  common  multiple  of  two  or  more  integral  expres- 
sions is  any  integral  expression  which  is  exactly  divisible 
by  each  of  them. 

The  lowest  common  multiple  (L.  C.  M.)  of  two  or  more 
integral  literal  expressions  is  the  integral  expression  of 
loivest  degree,  which  is  exactly  divisible  by  each  of  them. 


LOWEST  COMMON  MULTIPLE  159 

The  numeral  factor  of  the  L.  C.  M.  is  the  least  common 
multiple  (L.  C.  M.)  of  the  numeral  factors  of  the  given 
expressions. 

E.g.,  a^b^  is  the  L.  C.  M.  of  a^b,  ab^,  and  a^b^. 

Again,  the  L.  C.  M.  of  12  axy^z^  and  15  b^  is  60  ab^xy*^. 

169.   L.  C.  M.  by  factoring. 

Ex.  1.     Find  the  L.  C.  M.  of  aft^,  a^bc%  and  ab^c^. 

The  L.  C.  M.  of  these  expressions  cannot  contain  a  lower  power  of 
a  than  a^,  a  lower  power  of  b  than  b^,  and  a  lower  power  of  c  than  c^. 
Hence,  the  required  L.  CM.  is  a^bV'. 

Observe  that  the  power  of  each  base  in  the  L.  C.  M.  is 
the  highest  power  to  which  it  occurs  in  any  of  the  given 
expressions. 

When  the  expressions  involve  numeral  factors,  the  L.  C.  M. 
of  these  factors  should  be  obtained  as  in  Arithmetic. 

Ex.  2.  Find  the  L.  CM.  of  x^  +  7  x  +  12,  x^  +  6  x  +  8,  and 
5  x2  +  20  X  +  20. 

x2+7x  +  12  =  (x  +  3)(x  +  4); 
x2  +  6  X  +  8  =  (X  +  2)  (x  +  4) ; 
6x2  +  20x4-20  =  5(x  +  2)2. 

.-.  L.  C  M.  =  6(x  +  2)2(x  +  3)(x  +  4). 
These  examples  illustrate  the  following  rule  : 

To  obtain  the  L.  C.  M.  of  two  or  more  integral  expressions, 
multiply  the  L.  C.  M.  of  their  mimerol  factors  by  the  product  of 
all  their  prime  literal  factors,  each  to  the  highest  power  to  ivhich 
it  occurs  in  any  one  of  them. 

Proof.  The  L.  C.  M.  by  definition  contains  each  factor 
the  greatest  number  of  times  that  it  occurs  in  any  one  of 
the  given  expressions. 


160  ELEMENTS   OF  ALGEBRA 

Exercise  66. 
Find  the  L.  C.  M.  of  the  following  expressions : 

1.  4a^2/>  10a^2/^.  4.    a^,  x-  —  3x. 

2.  24a^6V,  eOa^dV.  5.    21  o^,  7ar>  +  l). 

3.  9  a'b^xy,  S  x^y^  6.    6a^- 2ii-,  Ga;^- 3  a;. 

7.  a^  +  2a;,  a;2^3^_^2. 

8.  aj2-5a;  +  4,  a;2_6a;  +  8. 

9.  a.-^  4- 4  a;  +  4,  a;2  +  5  a;  4- 6. 

10.  a^-a;-6,  a^  +  i»-2,  a;2_4^^3^ 

11.  a^  +  a;-20,  ar^-10a;4-24,  a;2-aj-30. 

12.  a;2_^a;-42,  ar^-lla;  +  30,  a^  +  2aj-35. 

13.  2a^  +  3.x'  +  l,  2a^+5a;4-2,  a^  +  3a;  +  2. 

14.  5a;2+lla;  +  2,  5a^  +  16a;4-3,  a^  +  5aj  +  6. 

15.  x^  —  7  xy  -\- 12  y^,  x^  —  6  xy  -{-  S  y^,  x^  —  5xy  -\-6  y^. 

16.  2a;2  +  3a;-2,  2a^  +  15a;-8,  a^  +  10a;4-16. 

17.  8a^-38x?/4-35/,  4  ar^  -  a;^/ -  5 /,  2x'-5xy-7f\ 

160.   L.C.  M.   by  H.  C.  F.     The   L.  C.  M.   of  two   expres- 
sions can  always  be  obtained  by  first  finding  their  H.  C.  F. 

Ex.  1.   Find  the  L.  C.  M.  ot  x^  +  x^  -  2  and  x^  +  2  a;2  _  3. 
The  H.  C.  F.  of  these  expressions  is  found  to  be  x  —  1. 
By  division  we  find  that 

JC3  +  ic2  -  2  =  (x  -  1)  (x2  +  2  X  +  2), 
and  x3  +  2x2-3  =  (x-l)(x2  +  3x  +  3). 

Since  x  —  1  is  the  H.  C.  F.  of  the  given  expressions,  their  second 
factors  x2  +  2  X  +  2  and  x2  +  3  x  +  2  have  no  common  factor. 
Hence,  the  required  L.  C.  M.  is 

(x-l)(x2  +  2x  +  2)(x2  +  3x  +  3).  (1) 


LOWEST  COMMON  MULTIPLE  161 

16L  To  find  the  L.  C.  M.  of  three  expressions  Af  B,  C, 
we  find  3f,  the  L.  C.  M.  of  A  and  B;  then  the  L.  C.  M.  of 

M  and  C  is  the  L.  C.  M.  required. 

Exercise  67. 
Find  the  H.  C.  F.  and  L.  C.  M.  of : 

1.  2a^  +  3a;-20,  6ic«-25a^+21a;-|-10. 

2.  a^-15aj-f36,  a^-3a^-2ar-|-6. 

3.  c)x^-x-2,  3a^-103^-7x-4:. 

4.  ix^  +  x'-Ax-i,  a^-{-6a^-\-llx  +  (i. 

5.  o^-x^-jx-^lo,  x^  +  x^-3x-^9. 

6.  a^-x^-\-x-\-3,  x'  +  a^-Sx^-x-{-2. 

7.  x*-x^-\-Sx-S,  x'  +  4:X^-Sx-{-2A. 

8.  6x^-{-x^-ox-2,  6x^-\-5a^-3x-2. 

9.  4:a^-10x^-{-4:X  +  2,  3a^ -2a^-3x-i-2. 

10.  x^-9xP-{-26x-24,  x'-Ua^  +  ATx-eO. 

11.  x^  —  ax^  —  a^x -\-  a%  x^+ax^~  a^x  —  a\ 

Find  the  L.  C.  M.  of : 

12.  x^-(ya^-\-llx-6,  ic^ - 9 o^  +  26 - 24, 

ar^_8a^  +  19a;-12. 

13.  a^-5x^+9x-9,  x^- x^-dx-\-9, 

x*-Ax'-^12x-9. 

162.  The  L.  C.  M.  of  two  integral  expressions  is  the  product 
of  either  expression  into  the  quotient  of  the  other  divided  by 
the  H.  C.  F.  of  the  two  expressions. 

Proof  Let  A  and  B  denote  any  two  integral  expressions, 
H  their  H.  C.  F.,  and  L  their  L.  C.  M. 


162  ELEMENTS  OF  ALGEBRA 

Then  L  by  definition  contains  all  the  factors  of  A,  and  in 
addition  all  the  factors  of  B  which  are  not  in  A ;  that  is, 
the  factors  oi  B-i-  H. 

Hence  L  =  Ax(B-^H).  (1) 

163.   From  (1)  in  §  162  we  obtain 

AxB=LxH.  (2) 

That  is,  the  product  of  two  integral  expressions  is  equal  to 
the  product  of  their  L.  C.  M.  and  their  H.  C.  F. 


CHAPTER   XII 
FRACTIONS 

164.  A  fraction  being  an  indicated  quotient,  the  fraction 
a/6  denotes  that  number  which  multiplied  by  the  divisor 
b  is  equal  to  the  dividend  a.     Reread  §  90. 

E.g.,  -  8/4  =  -  2,  for  -  2  x  4  =  -  8. 

165.  Algebraic  fractions.  The  fractions  in  arithmetic  in- 
volve only  arithmetic  numbers,  and  are  called  arithmetic 
fractions. 

In  Chapter  III.  we  used  arithmetic  fractions  to  denote 
the  arithmetic  values  of  positive  and  negative  numbers,  the 
quality  being  indicated  by  the  sign  ■•"  or  ~. 

An  algebraic  fraction  is  one  whose  numerator  and  denomi- 
nator are  quality-numbers.  The  sign  before  an  algebraic 
fraction   denotes   the    quality   of    its    numeral   coefficient. 

Thus,  —  ^^^  denotes  the  product  of  —  1  and  the  fraction 

(-4)/(+3). 

166.  By  the  law  of  quality  in  division  it  follows  that  — 

Changing  the  quality  of  both  the  nuynerator  and  denominator 
does  not  change  the  quality  of  the  fraction. 

Changing  the  quality  of  either  the  numerator  or  denominator 
changes  the  quality  of  the  fraction. 

^*  -8     8'  h        -b 

But  ^^  and  -^^  are  opposite  in  quality. 
6  —  b 

163 


- 

-8 
9 

8 
~9' 

«_ 

—  a 
b 

or  -- 

a 
-b 

abc_ 
xyz~ 

,(z 

a)(-6)(- 
xyz 

ic). 

—  abc 
xyz 

=  - 

(- 

-  abc 
-x)yz 

164  ELEMENTS  OF  ALGEBRA 

167.  By  §  166  and  the  law  of  quality  in  §  48  it  follows 
that  — 

Changing  the  sign  before  a  fraction  and  changing  the 
quality  of  either  its  nuynerator  or  denominator  does  not 
change  the  quality  of  the  term. 

E.g., 

Again, 

168.  A  fractional  literal  expression  is  an  expression  which 
has  one  or  more  fractional  literal  terms. 

E.g.,     and  ax  +  by  -\ are  fractional  expressions. 

x-y  a+b 

An  integral  literal  expression,  as  we  have  seen,  denotes 
any  integral  or  fractional  number;  likewise  a  fractional 
literal  expression  denotes  any  integral  or  fractional  number. 

E.g.,  a  denotes  2,  5,  3/2,  —  2/3,  or  any  other  number. 
Again,  when  a  =    6  and  6  =  2,  a/b  =  3 ; 

when  a  =  12  and  6  =  3,  a/b  =  4  ; 

when  a  =    3  and  6  =  5,  a/b  =  3/5  ;  and  so  on. 

169.  A  proper  literal  fraction  is  a  fraction  whose  numer- 
ator is  of  a  lower  degree  than  its  denominator  in  a  common 
letter  of  arrangement. 

An  improper  literal  fraction  is  a  fraction  whose  numerator 
is  of  the  same  or  of  a  higher  degree  than  its  denominator  in 
a  common  letter  of  arrangement. 

1  J*  4-  1 

E.g.,    — = —  and "-^-^ are  wroper  literal  fractions. 

While    — - —  and  — ^— t are  improper  literal  fractions. 

x-2  +  1  x2  +  3x-4 

The  value  of  a  proper  literal  fraction  may  be  either  less  or  greater 
than  1. 


FRACTIONS  166 


REDUCTION  OF  FRACTIONS. 

170.  To  reduce  an  expression  is  to  find  an  identical  expres- 
sion of  some  required  form. 

171.  To  reduce  an  improper  fraction  to  an  expression  con- 
taining no  improper  fractions, 

Perform  the  indicated  operation  of  division. 

Sometimes  the  quotient  can  be  obtained  by  inspection. 
For  examples,  see  §§  129  and  133. 

Exercise  68. 

Reduce  each  of  the  following  improper  fractions  to  frac- 
tional expressions  containing  no  improper  fractions : 

,     5ar^-20a;-15      ^     -^    ,  ^  o 

1.   •     0.   •  y. 


ox 

x  +  7 
x  +  2 

x  +  3* 
5x-\-7 


x'  +  a' 

x-\-a 

ar^  +  a'^ 

x  —  a 

^-a' 

X  +  a 

a^  +  16 

10. 


11. 


12. 


x*  + 

a' 

X  — 

a 

2x2 

-7x 

-1 

x-3 

a^- 

■3x 

X- 

-2 

Sor 

+  2x 

4-1 

x-3  x  +  2  x-\-4: 

4 g^  +  6 a6  +  9  6^  ^^     60x^-17x^-4.-^  +  1 

2a-3h        '  '  5x2  +  9x-2 

172.    The  value  of  a  fraction  is  not  changed  by  multiplying 
its  numerator  and  denominator  by  the  same  number. 

That  is,  a/b  =  am/(bm). 

Proof  -  =  -  X  -,  m/m  being  a  form  of  1 

b      b     m 

=  am/(bm).  §  91 

■^■'    4     4x6     20'  x+y     (x  +  y)(x  +  y)      (x-Vy)'^' 


166  ELEMENTS   OF  ALGEBRA 

173.    The  value  of  a  fraction  is  7iot  changed  by  dividing  its 
numerator  and  denominator  by  the  same  number. 

That  is,  a/b  =  (a^  m)/(b  ^  m). 

Proof  ^^'^^  =  (^^^^0^^^  =  ±  §  172 


E.g., 


b  -i-m      (b  -i-  m)  m      b 

c  -\-  ex  _{c  +  ex)  -4-  c _  1  ■\-x 
G  +  cy~  (c  -\-  cy)  -^  c~  1  -^  y 


174.  A  fraction  is  said  to  be  in  its  lowest  terms  Avhen  its 
numetator  and  denominator  have  no  common  factor. 

175.  To  reduce  a  fraction  to  its  lowest  terms, 

Divide  its  numerator  and  denominator  by  all  their  common 
factors,  or  by  their  H.  C.  F.  (§  173). 

Ex.  1.    Reduce     ^^  ^    to  its  lowest  terms. 
8  a'^xy^ 

The  H.  C.  F.  of  the  numerator  and  denominator  is  4  axy^  ;  and 

4  ax^j^  _  4  ax^y^  -^  4  axy^  _    x^ 
8  a%y*  ~  8  a^xy^  -^  4  axy^  ~  2  ay^' 

Ex.  2.    Reduce  £JZ^  to  its  lowest  terms. 

Factoring  numerator  and  denominator,  we  obtain 

a'^  —  ax_       a(a  —  x)       _     a 
cfi  —  x'^~ {a  ->t  x){a  —  x)~ a  -\-  X 


173 


§173 


Ex.  3. 


X*  - 1  _  x^-\ 


x^  —  x^  —  x"^  -{-  X     ic2(a:4  —  1)  —  x(x*  —  1) 
_      1      _        1 

~  x^  —  x~x{x  —  1) 

Exercise  69. 
Reduce  to  its  lowest  terms  each  of  the  following  fractions : 

■    -ajy  ■  4aWc 


FRACTIONS 


167 


xY^"^ 


3- r7-2- 

—  5  ab^xifz 
5  a^b*<^x}f 

g     Sa^bc*x^fz 
4  a^b^ca^y^z^ 

10.  «;-°^ 

a''*  +  a6 

12.  ^-^y". 

_.     a^  +  2a; 

,.     4.T-16 

15     2ar^-4a;\ 
ar*  —  4  a;'' 


17. 


20. 


a-3 
9-a- 


18.     ^'-^'. 

0^—1 


a;'*  —  a* 
x^  —  a^ 


16. 


x-2 
4.-x^' 


15  g^  —  5  aa; 

g^  -  2  oa;  +  a^ 
ar^-g'' 

g^  +  2g^6^  +  6^ 
61 -g* 

l-5g  +  6g» 

•  l_7g  +  12g2 

jr  -\-6x  —  55 

l-9f-^20y* 

•  1  +  62/^-552^ 

a:^  +  :r^-2 
2g     a;'^  +  2a;>'  +  l 

(a^+f)(x'+xy+y') 

-     ar^  —  ga.*^  +  6^a;  —  ab^ 
a^  —  aji^  —  b^x  +  g6- 


168  ELEMENTS   OF  ALGEBRA 

176.  When  the  factors  of  the  numerator  and  denominator 
of  a  fraction  cannot  be  found  by  inspection,  their  H.  C.  F. 
can  be  found  by  the  method  given  in  Chapter  XI. 

Ex.  1.   Reduce    ^^^  ~  '^^^^^  23  a; -21   ^^  ^^  lowest  terms. 
15a;3- 38x^ -2x  +  21 

The  H.  C.  r.  is  found  to  be  3  a;  —  7  ;  and  by  division  we  find 

3x3  -  13  x2  +  23x  -  21  =  (x2  -  2 X  +  S)(Sx  -  7) 

15x3  -  38x2  -  2x  +  21  =  (5x2  -  X  -  3)(3x  -  7). 

3  x3  -  13  x2  +  23  X  -  21  _  x2  -  2  X  +  3 
"  15x3 -38x2 -2x  + 21"  5x2 -X- 3' 

Exercise  70. 
Reduce  to  the  simplest  form  the  following  fractions: 
a3_3a-f-2  ^     2  0?  ^  cux" -\- 4.  a^x  -  1  a^ 

!•    - — :; :: — 7. r'  "■ 


•     a4  _  ^2  _  12 

o.     — : r :•  o* 


2a^-3a2 

+  1 

a«  +  3a2- 

-20 

a>-a^- 

12 

4  x^  +  3  aa;2  +  a 

3 

x''-^ax'-\- 

a^x^ 

a* 

4a^-10a; 

2  + 4a 

•  +  2 

^x'^-2y? 

-3aj 

+  2 

6a;^  +  a^- 

-5a;- 

-2 

a^_7aa^4_8a2a'_2a3 

2a^  +  3a^  +  4a;-3 

6a^  +  a^_l 

a;4  _  a^  _  a;  _|_  1 

a,4_2a^-a^-2aj4-l 

4a;*  +  lla^  +  25 

4a;*-9a^-f30a;-25 

i»4-20a.-2-15x  +  4 

^*   6a^-i-5a;2-3x-2*  "    x'*-f9a^+19ar'-9a;-20 

177.  Two  or  more  fractions  which  have  the  same  denomi- 
nator are  said  to  have  a  common  denominator. 

The  lowest  common  denominator  (L.  C.  D.)  of  two  or  more 
fractions  is  the  L.  C.  M.  of  their  denominators. 

E.g.,    the    L.  C.  D.    of    the    fractions    — -^    and    — ^-^— -    is 

(«  _  5)2^(j  ^  ft)^  or  the  L.  C.  M.  of  the  denominators  ol^  —  IP-  and 
(a  -  6)2. 


FRACTIONS  169 

178.  To  reduce  two  or  more  fractions  to  identical  fractions 
having  the  L.  C.  D., 

Multiply  both  the  numerator  and  the  denominator  of  each 
fraction  by  the  quotient  obtained  by  dividing  their  L.  C.  D.  by 
the  denominator  of  thai  fraction. 

Proof  The  derived  fractions  have  the  L.  C.  D. ;  and  by 
§  172,  each  is  identical  with  its  corresponding  given  fraction. 

Ex.  1.   Reduce ,  — — -^ ,  and  - — -^ to  iden- 

a^h{x  +  a)    ah-\x-a)  ah{x^  -  d^) 

tical  fractions  having  the  L.  C.  D. 

The  L.  C.  M.  of  the  denominators  is  a'^b^{x^  -  a*). 
Dividing  this  L.  C.  D.  by  the  denominator  of  each  fraction,  and 
multiplying  both  the  numerator  and  denominator  by  the  quotient, 
we  obtain 

X         _xxh(x  —  a)_    bx(x  —  a) 
a26(x  +  a)  ~  a-^62(a;-2  _  a^)  "  a^ft^Cx^  -  a^y 

y        -y  X  a(x  +  a)_    ay(x  +  a) 
ab\x  -  a)  ~  a^b'\x^  -  a«)     a^b^x:^  -  a'^) ' 


and 


_      z  X  ab       _         abz 


abix:''  -  a?')     a'^b\x^  -  ««)     a^b^x^  -  «^) 


Ex.  2.   Reduce   — — ;? -,   — — ^ -,   — — to  iden- 

««  -  5 a;  +  6    x^-ix-\-S    a;^  -  3 x  +  2 

tical  fractions  having  the  L.  C.  D. 

The  denominators  equal 

(x_3)(x-2),  (x-3)(x-l),  (x-2)(:«-l), 

respectively.    Hence  their  L.C.  M.  is  (x  —  3)(x  —  2)(x  —  1). 

1  _  x-1 


{X- 

-3)(x- 

-2)      (X  - 

-3)(x- 

-2)(x- 

-1) 

1 

X  - 

-2 

c^- 

-3)(x- 

-l)-(x- 

-3)(x- 

-2)(x- 

-1) 

1 

X  - 

-3 

(x-2)(x-l)      (x-3)(x-2)(x- 1) 


170  ELEMENTS   OF  ALGEBRA 

Exercise  71. 
Eeduce  to  identical  fractions  having  the  L.  C.  D. : 
,      3        4         5  ^234 


Ax    6a^'    12a^  *  a  -  b'    a  +  b'    a'  +  b' 

5a^      Sbx     1  cy  —  m  ay  ax^          xy^ 

Q>x^y     Sfx       10 xz'  '  r^'  (1-a;/  (1-xf 

a                X                O?  a  '^             '"'^                  '*"^^ 

^'    J    • >     — -»'  *>•  —y 

X  —  ax  —  ax-  —  a^ 

ab  m  —  n 


7. 


am  —  bm  -\-  an  —  bn    2o?  —  2  ab 
3  5  2 


a^  +  3u;  +  2'   a;2  +  2a;-3'   :f?^hx-^^ 
^       2  a  b  3a^  5  6^ 


10. 


a-b'   2b-2(K   4.(a'-b'')'   6(b'-a') 
1  1  1 


(x  —  a){x  —  by    (b  —  x)  (c  —  ic)'    {x  —  c)(x  —  a) 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS. 
179.   The  converse  of  the  distributive  law  for  division  is 

a      b      c  _a-\-b  —  c 
x     x     x~        x 

Hence  to  add  or  subtract  fractions, 

Reduce  the  fractions,  if  they  have  not  a  C.  D.,  to  identical 
fractions  having  the  L.  C.  D. ;  then  add  or  subtract  each  nu- 
merator as  the  sign  before  the  fractiori  directs,  and  write  the 
result  over  the  L.  C.  D. 

h     c__hy     cx_by  +  cx 


and 


X     y     xy     xy         xy 

a     b  _ax     bc_ax  —  be 
c     x"  ex     cx~      ex 


FRACTIONS  171 

Note.  The  student  should  remember  that  when  either  the  numer- 
ator or  the  denominator  is  a  polynomial,  the  horizontal  line  in  a  frac- 
tion is  a  sign  of  grouping  as  well  as  a  sign  of  division. 

Ex.  1.   Combine  and  simplify 1- 


The  L.  C.  M.  of  the  denominators  is  {x  —  y)(x-\-  y);  and 

x-y     x-\-y     (x-y)(a:  +  y)      (x-y){x  +  y) 
_.a;  +  y-Ka;-y)_     2x 

1  1 


Ex.  2.   Combine  and  simplify 


a;2-5x+    6  =  (a;-2)(x-3), 

x^~lx^  12  =  (x-3)(x-4); 

hence,  the  L.  C.  M.  of  the  denominators  is  (x  —  2)(x  —  3)(a;  —  4),  and 

the  expression  = ^-^^^ ^  ~   

(x_2)(x-3)(x-4)      (x-2)(x-3)(x-4) 

_      (x-4)-(x-2) -2 

(X  -  2)(x  -  3)(x  -  4)  -  (X  -  2)(x  -  3)(x  -  4)' 

Ex.  3.   Combine  and  simplify  g^ZL^  _  «£_i^^  _  «fe  -  c^ 

he  ca  ah 

The  L.  C.  M.  of  the  denominators  is  ahc,  hence 

the  expression  =«(«1:^M _  Hac~b')  _ c(ab  -  c^) 
dbc  abc  abc 

_,a(a^  -  be)-  b(ac  -  b^)-  c(ab  -  c^) 
~  abc 

_fl^  +  b^  +  c^-Sabc 
~  abc 

Exercise  72. 

Combine  and  simplify  : 

,     o,  —  5b     a  — 3  b                   „     a  —  36,3a  —  6 
1. ^' 1 z — 


172  ELEMENTS  OF  ALGEBRA 


6  a  —  5  b     4  a  —  76  2^~'^      q^  +  ?/ 

3  2       '  '3  4    * 

_     X     x  —  4:,x  —  5  ^a  —  b      „a  — 6 

5-    T 3 1 —  8.    5 7 

4         3b  ex 

2.T-3     a;4-2     5a;  +  8 
9  6     "^     12     ' 

10  g  — 26     a-5  6«  +  7& 

2  a  4  a  8  a 

11  b  -\-  c  .  c-{-a     a  —  b 

2a         Ab  3c 

a  —  a;a4-ic     a^  ~  a^ 
14. 1- 


13. 


X  a  2  ax 

2  gg  -  62         62  _  g2         ^2  _  ^2 

a'-^  6^  c2 


^^     2a;-3y     3a;-2g     5 

xy  XZ  X 

15     a;-3a;^-9      8-a^ 
5a;         10a;2        15a^'^ 

16.  l-^Mn^  +  ^^^hl!. 

a;?/  xy^  x^y^ 

Keduce  to  an  improper  fraction : 

.1.      a^  +  6^ 

17.  a  +  6 -^—  • 

a  —  6 

q  +  6     a2_^ft2_^2_  52_(^2^52) 
1  a  —  b  ~  a  —  b 

~a  —  6~6  —  a 

18.  a-l4--4-T-  20.   a;4-2?/4--^^- 

a  4- 1  X  —  2y 

19.  a  +  a;+-^.  21.    x^  _  3x  -  ^^^?^=^. 

a  —  a;  a;  —  2 


FRACTIONS  173 

Gar* 


22.    a^  — 2 ax +  4:0^  — 


23.   x-a  +  y  +  '^'-^'y-^y" 


a-^2x 

HA 

x-\-  a 


24.  l  +  aj  +  ic'  +  a^'  +  r-^- 

1  —  a; 

25.  ^;+2a^  +  l_  _ 

26.  l  +  2a;  +  4a52  +  -^-+i. 

2ic  —  1 

ar^  +  13a;-5 


27.    cc2_2a;  +  3- 


28.   Combine  and  simplify  — ^  +     ^^ 


a  —  ic     a^  —  a* 

Beginners  should  always  see  to  it  that  the  denominators  of  the 
fractions  to  be  added  or  subtracted  are  all  arranged  in  descending 
powers,  or  all  in  ascending  powers,  of  some  particular  letter  of 
arrangement. 

Arranging  the  denominators  in  this  example  in  descending  powers 
of  a,  we  have 

a  ax     _    a      ,     — «« 


a  —  X     x^  -  a^     a  —  X     a^  —  x^ 

_a(a  +  x)  —  ax  _      a^ 
-       rt2  _  a;'2        "  a2  -  x^' 

Combine  and  simplify : 

29.  -^  +  -*-.  33.    ^-  +  -1-. 
a  —  b      b  —  a  3  +  x     a^—d 

30.  .-^_  +  -«-.  34.    l+?-ll^. 
x  —  aa  —  x  1  —  ojl-fa; 

31.  '^       +       «      .  36.    ^-  +  — ? 

32.  -^ 2_.  3       a  +  2b_a-2b_ 

1-x     1-x'  a-2b     a  +  26 


174  ELEMENTS   OF  ALGEBRA 


x-\-y      x  —  y  '    4y^ 

x  —  y      x^y  xr  —  y^ 

38.  2a     ^     26  ,C' ^h\ 
a-j-b     a—b  a?  —  b^ 

39.  -^ 1-  1 


a  —  1  a  (a  —  1) 

4.o?-b^     2a-[-b 
41.    -^+     1  1 


a;  — 1      ic  — 2     ic4-2     x-\-l 

The  character  of  the  denominators  in  this  example  suggests  that  it 
is  simpler  first  to  combine  the  first  and  fourth  fractions,  next  the 
second  and  third,  and  then  to  combine  these  results,  as  below : 

1  1     _x  +  l  -  (r<;-l)_     2 


x-\     x-\-\  a;2  _  1  a;2  -  r 

^ 1     _a;  +  2-(a;-2)_      4 

X  -  2      a;  +  2 "  a:2  _  4  "a;'^  _  4' 

and  ^       .        4     ^ 2(a;2-4)+4(x2-l)  ^    6x^-12 

x2-l     x2-4~      (a;2_  i)(a;2_4)      "a:*- 6x2  + 4' 

42.    ^  +  ^?_+     2     ^.     4 


1— ic      l  +  a;      1  +  a^      l  +  ic* 

Here  it  is  simpler  first  to  combine  the  first  and  second  fractions, 
next  to  combine  this  result  and  the  third  fraction,  then  this  last  result 
and  the  fourth  fraction,  as  below  : 


2 ' 


1-X        1+X  1-X2  1-X 

2  2     _2(1 +x2)+2(l -x2)_      4 

1-X2  1  +  X2~  l-X*  ~1-X* 

4  4     _4(1 +  x*)+4(l -x4)^      8 

1  _  X*  1  +  X4~  1  -  X8  1  -  XS" 


43. 


FRACTIONS  175 

2  1  ic  +  6 


x-2     x-\-2     ar^  +  4 
44.    -^  +  _iL_+     2a^ 


a  —  ic  a  +  a;      a^  +  a^ 

45      3 -a;  3  +  a;       l-16a? 

l-3a;  l+3ic      9x2_i' 

1  1               a.'  +  3 


46. 


x-1      2(a;  +  l)      2(ar'  +  l) 


.^        a      ,      a      ,      2a2     ,     4  a* 

47. 1 r-^— — n-T—r-, — i* 

a  —  X     a-\-x     a-  -f  ar     a*  -f  a;* 

48.    ^-^+      3  1 


a;  —  3     a;  — 1     a;  +  1     a;-|-3 
49.        1      -_±,+6__4_^     1 


a;  — 2     x  —  1      X     x-{-l      x-\-2 
50.      ,      f        ,+  2  1 


aj2_3a;  +  2     ar^-a;-2     ar'-l 
The  expression 


51. 


52. 


53. 


54. 


(a;_2)(a;-l)      (a;-2)(x+l)      (x-l){x  +  \) 
2(x  +  l)+2(a;-l)-(a;-2) 
(x-2)(x-l)(a;+l) 
3  X  +  2 
(x-2)(x-l)(x+l)* 

1.1 


x^-dx-[-20     xr-llx-\-30 
1  1 


a^- 

7x 

4-12 

a;2- 

-  5a; 

+  6 

1 

1 

2ar^ 

—  X 

-1 

2x2^3.. 

-3 

1 

3 

2aj2 

—  X 

-1 

6ar^ 

—  X- 

-2 

4 

3 

55. 

4-7a-2a2     3-a-lOa" 


176  ELEMENTS   OF  ALGEBRA 

56.    ? ? 

5  +  x-lS^     2-h5x  +  2x' 

g^  5x 15(a;-l) 9(x-\-S) 

2(x-\-l)(x-3)      16(x-3)(x-2)      16  (a; +1)  (a; -2)' 

58.   ^^^+  1^  12 


x^-{-5x  +  6     ar^  +  9ic  +  14     a^H-10a;4-21 
69.   _^  +  -A_+       40.4-2 


a^-1      2a;  +  l      2a^  +  3ic  +  l 

24  a; 34-2a;     3-2a; 

•    9_i2a;4-4a;2     3-2a;     3  +  2a;' 

61.  1  2^1 

a;^  4-  5  aa;  +  6  a^     ar^  -f  4  aa;  +  3  a^     a;^  4-  3  ax  -\-2a^ 

62.  —^ "'-"  +       2       ■ 

(a;  —  2  a)^     a^  —  5  aa;  4-  6  a^     a;  —  3  a 

63.  1_     4     ^     6      _     4     ^      1 


a     a4-l      a  +  2     a -^3     a  4-4 

1  2  1 

64. — :--, : — :4- 


a^_5a;  +  6     x'-4,x-\-3     '.x^-3x  +  2 
65.    ,A^-;,-V+       ^ 


8_8x     8  4-8a;     4  4-4a.'2     24-2a;* 

1  1  1       ,       18 

66. tt:-:; -tt:  -    o  .   ..  + 


6a-18     6a4-18     a^  4-9     a^  4-81 
6T.     ,         1  .  +  ,.         ,^         ,+  1 


(a  —  b)(a  —  c)      (b  —  c)  (6  —  a)      (c  —  a)  (c  —  5) 

In  examples  of  this  kind  it  is  best  for  beginners  to  arrange  all  the 
factors  in  the  denominators  of  the  fractions  so  that  a  precedes  b  or  c, 
and  b  precedes  c. 

We  therefore  change  b  —  a  into  —  («  —  6),  c  —  a  into  —  (a  —  c), 
and  c  —  b  into  —(b  —  c).     The  expression  then  becomes 

1 1 + 1 

(a  -  &)  («  —  c)      (a  -  &)  (6  -  c)      (a  -  c)  (6  -  c) 


FRACTIONS  177 

The  L.  C.  M,  of  the  denominators  is  (a  —  b)(a  —  c)  (6  —  c) ; 

•••  the  expressions    (a  -  6)(a  -  c)(6  -  c)    -^- 
68.  c  .  a  .  b 


91 


(6-c)(6-a)      (c-d)(c-b)      (a-b)(a-c) 

69.    ? + - + ^ 

(y-x)(z-x)      (y-z){y-x)      (z-x){z-y) 

70.  y  +  z  I  z  +  x  ^  x-\-y 
{y-x)(z-x)      (y-z)(y-x)      (z-x){z-y) 


MULTIPLICATION  AND  DIVISION  OF  FRACTIONS. 

180.  Product  of  fractions.     See  §  91. 

Ex    1     a;  +  2  ^  X  +  3  ^^  X  +  4  _  (a:  +  2)  (x  +  3)  (x  +  4)  _  ^ 
■x  +  3     x  +  4     x  +  2      (x  +  3)(x  +  4)(x  +  2) 

Ex.  2.    Simplify  ^^-^^  x  -M±J/L.  x  -^M^ 

The  factors  common  to  numerator  and  denominator  can  be  can- 
celled before  the  multiplication  is  performed,  as  below  : 

The  expression  =^(2-ri)  x  ^(^^J^   ^ W 

_       2x 
~3(x  +  2j/)" 

181.  To  multiply  a  fraction  by  any  number,  , 

Multiply  the  numerator^  or  divide  the  denominator,  by  that 
number. 

Proof.  ^xm^^X?^^  §91 

0  0      1        b 


=  a/(b-^m).  §173 

8  62"  8  62      "  8  62^4' 


Ex.    ^x4=^«iAior      ^«' 


182.    The  reciprocal  of  a  fraction  is  equal  to  the  fraction 
inverted. 

That  is,  1  -J-  {a/b)  =  b/a. 


178  ELEMENTS   OF  ALGEBRA 

Proof,     b/a  multiplied  by  the  divisor  a/b  is  equal  to  the 
dividend  1 ;  hence  b/a  is  the  quotient. 

183.    To  divide  by  a  fraction, 

Multiply  by  the  reciprocal  of  the  fraction. 

Proof.     Dividing  by  a  number  gives  the  same  result  as 
multiplying  by  its  reciprocal  (§  87). 

Ex    1     3«  ■  ^x_Sa  ^^7  y_2lay 
*    56  ■  ly~bh      2x~106x' 

Ex    2      ^  —  «    .  x^  —  a^  _  X  —  a        x  +  a 


x^  +  a^      x  +  a      x^  -^  a^     x^  —  a^ 

\ 


(x2  -  ax  +  a^)  (ofi  -\-ax-^  a^) 
1 


a;4  +  a'^x'^  +  a* 

184.   To  divide  a  fraction  by  any  number, 

Divide  the  numerator,  or  multiply  the  denominator,  by  that 
number. 

Proof  «^m  =  -x-  =  —  §91 

b  b      m      bm 

=  (a-^m)/b.  §173 

Exercise  73. 
Simplify  each  of  the  following  expressions : 

1.    l^x'p-  6.    !^X^-^'- 

3 c     4a  a     c      a 

^     2a   .6  c     5x  ^     G?   .^     o. 


"'    3b^  ha^2y  ^'    b^^  f  '  / 

^     2a     5c     x'b  ^     Sa^  ^   2c        Sa 

o.  X  —  X *  o.     — — —  X 


5b       X        y  4.b      7ax     7bx^ 

^     2  a'  .  Sabx  ^  5f^^^21c,  .  35  (fy 

be         c'y  '  7  a^     4:  ax      7  a^x 

3axy  ^  6aV  -^  2  b  ,  x     9a^ 

'     56^    '  lOba^'  '  3a'  V     4  6^* 


17. 

x-2'' 

x  —  2      X- 
X  —  S  '  X  - 

-4 
-3' 

18. 

a^-a' 

x  +  2a^ 

ar-4a 

^       x  —  a 

19. 

a'  +  a^ 

Aa-x)\ 

9ft 

Ux"- 

7x       2x- 

1 

FRACTIONS  179 


x^  -\-xy     xy  —  y^ 
_-     a^  +  2a;,,g^-3a; 

XT  —  ^y^       X  -\-  y 

14     _a±h_     ab--^ 

a^-a'b     ab  +  a^  "    12a^  +  24:x'  '  x" +  2x 

^g     ar>  +  3a;^  .    x  +  S  ^^     16a^-9a^  .  4a;-3a 

x  +  4     '  x?-\-4:X  '       «2-4       '     a;-2 

,.      a  +  46    .   ah-^AW  ._     a2/>2-|.3a&     aft  +  3 

(ji-^hab      a3  +  5a-6  4a2-l        2a+l 

ar'-14a;-15  .  a^-12a;-45 

•  a:2_4a._i5    '    ar^_6a;-27* 

24     ar^-6a.-^  +  36a;  .    a^  +  216a; 
iB2  _  49  '  Q^-x-^2 

y?-x-2^  ,    x-\-\     .a^H-2a;-8 
ar'-25      '  or^-h  5a;  '    ar^  -  a;- 2  * 

ar'-lSx  +  SO  .  a;^-15a;  +  56     a;4-5 

•  ar*_5a;_50   *     ^-^x-1       x-l 

27  af^-8a;-9        a;^  -  25  .  a;^  +  4a;  -  5 
^-Vlx^l2       x^-l    '  a^-9x  +  S 

28  a;^-8a;     ^ar^  +  2a;  +  l   .a.-'  +  2a;  +  4^ 
a;^  —  4a;  —  5     ar'  —  a;-  —  2a;*        x  —  5 


29     (g  +  sy  .    g-^-ft^     .  (a-\-by 
'    (a  -  6)»  *  (g2  -  62)2  •    ^2  _^  ^2 

.    (a  -  cy  -b'-  '  b^-(c-  ay 


180  ELEMENTS   OF  ALGEBRA 


31.    ^^  +  /  X  ^  ~  ^  •  ^^  ~  ^''^^  "^  ^*- 
'   QC^  —  y^     x-\-y  '  X*  -\-  x^y^  +  y^ 

First  reduce  each  of  the  mixed  expressions  to  fractions. 
33.    (a  +  ^!^\(b         «'' 


a—  bj\        a  +  6 


34.    of_±xy^r_x_ y\ 

\x  —  y     x-\-y) 


^  -\-  y^     \^  —  y    ^  +  Vy 

35     fci  +  ba  —  b'\^/'a-{-b      a  —  b\ 
\a  —  b     a  -j-  bj  '  \a  —  b     a  -\-  bj 

"■(-¥)x(3^-')*J 

2  a  (a  —  by  \        ctj      \       «/ 

38     4a^  +  a;-14  ^    4fl;^    ^    a;  -  2  _^     2a^  +  4fl; 
6  a;?/  —  14  2/       a^  —  4     4fl;  —  7*3ar^  —  a;  —  14 

x^  +  a;-2       a^2  +  5a;  +  4  .  /a;^  +  3a;  +  2  .^-{-3 
•    aj2_^._2o'^       or^-o;        *  l^a.-' -  2  a^  -  15         a^   J 

._     4a^-16a;  +  15         a;^-6a;-7      ^,         4ar'-l 


)• 


2aj2^3a;  +  l        2a;2_l7a;  +  21     4a^-20a;  +  25 
a^  4-  a6  —  ac      (a  +  c)^  —  6^     ab  —  b^  —  be 

a2-62_c2_26c      62_2?,c  +  c2-a** 

43  a;^-64  ar'-H2a;-64  .  a?^-16a;  +  64 

ic2  4-24a;4-128  aj3_64        •    a^^4,x-^16' 


FRACTIONS  181 

185.   A  complex  fraction  is  a  fraction  whose  numerator  and 
denominator,  either  or  both,  are  fractional  expressions. 

g  +  6 

E.g.^  ,  or /(-  +  -),  is  a  complex  fraction. 

^  _L  £         c  —  d/\x     yl  ^ 

X      y 

Observe  that  a  heavy  line  is  drawn  between  the  numerator  and 
denominator  of  the  complex  fraction. 

ay 
bx 


Ex.1.  ^/^=^xy= 

b/  y      b     z 


Ex.  2. 


+  x 


X 


x+j,^_x_ 


y       y  +  x    y 

Sometimes  the  easiest  way  to  simplify  a  complex  fraction 
is  to  multiply  its  numerator  and  denominator  by  the  L.  C.  M. 
of  the  denominators  of  their  fractional  terms. 


a-\-x     a  —  x        a-\-x     a  —  x\,  ..     , 

; —        — ; —    (a  -  x)  (a  + 

a  —  x     a -{- X      \a—x     a  +  x/ 


._..     „^..       .„_..      .^...  ^) 

Ex.  3.  _  , 

a  +  xa-x      (a-\-xa-x\.  ..     ,     , 
h — ; — h — ; — )  (a  -  x)  (a -{- x) 

a  —  x     a  +  x      \a-x     a-\-xJ^  ^^  ^ 

-  (a  +  x)2  +  (a-a:)2 

_    2  ax 

~  a2  -1-  x2* 

Here  (a-x)(a  +  x)  is  the  L.  C.  M.  of  the  denominators  of  the 
fractional  terms  in  the  numerator  and  denominator  of  the  complex 
fraction. 

^^4  X  _\        X  J     _x*  +  a-x^  _      x2 


Ex.  5.  Simplify  


x  +  2 


x  +  2 
X4-  1 


182  ELEMENTS   OF  ALGEBRA 

In  a  fraction  of  this  kind,  called  a  continued  fraction^  we  first  sim- 
plify the  lowest  complex  fraction  as  below  : 

X X 

X4-2  =^  {x  +  'l)x  §83 


^  _,  2      ^  +  1  (a;+2)x-(a:+l) 


X 


x^  +  2x 
x^-\-x-l 
xCx"^  -h  X-  I) 


X  (x'^  +  X  -  1)  -  (ic2  +  2  x) 


Exercise  74. 
Simplify  each  of  the  following  fractional  expressions: 

3a  +  —  -  +  - 

8c  ^     b      d 

4.    ^r-r'  7. 

Q      ,76 
3c+^ 

8a 

5.  ^. 


1. 

71      m 

a      6 

m      n 

2. 

x-y 

d 

3. 

46 
"+3 

10. 

12      3 

^—x 

X 

11. 

2x'-x-6 

S-' 

X 8. 


m 

X 

_1 

a; 

l+i 

a; 


ar  X 

a;^  X     OCT 


a  -\-b      a  —  b 


a 
12.    - 


^  _  a^  +  6^ 


13. 


(a  +  6)^ 
x^'^a' 
a^     ax     x^ 


FRACTIONS 


183 


14.    1 -f 


l-\-x 


2x2 


15. 


16. 


a  — 


a  + 


17. 


18. 


19. 


1- 


1  +a; 

1 

X 

X 


x-2- 


X  — 


x-1 

x-2 
1 


x-\- 


m 


2/4- 


ic  + 


x  + 


20. 


21. 


22. 


23. 


24. 


25. 


x  —  y  — 


x-y-\- 


xy 


x-y 


x  +  y 


x-\-y 


xy 


x  +  y 
l-x  ,  l  +  2aj2l 


1         l-fo:       1 


a;-2 


1 


a;-2 


a;-4 


X 

01?  — y^ 

^   ~ 9 

x--y^ 

x-\-l 
2  0^  +  1* 

4 


ic  —  4 


x-2 


X  —  b 


X—4:  — 


X—  4: 


a"  +  h^ 
2  ah 


^\     aW      .  4a6(a-f  ?>)  ' 

|i  +  _^+_-^Ui i_l 


V      ict  +  by] 


1  + 


a  +  6 


^^*  U     W"^S  A2/"^W     ^/ 


184 


27. 


28. 


29. 


30. 


6« 


ELEMENTS   OF  ALGEBRA 

a-b       a'-  a^b'  +  b* 


a^-W 


a' 


b'  '  a'  +  a'b'  +  b' 


x  +  2       4aH-5\     /2  a;  +  3     3  a;  +  4 


2aj  +  3     5x-{-Q>]     V3i»  +  4     4.x-{-6 
l-\-x      l^a?\     fl-\-x^     1-^x 


1  +  a?      l  +  xV     Kl  +  x"     l-\-x' 


a  -  by 

a  +  b 


a  +  b 


a  —  bj         a  —  b        J 


186.  Power  of  a  fraction.  The  nth  power  of  a  fraction  is 
equal  to  the  nth  power  of  its  numerator  divided  by  the  nth 
power  of  its  denominator ;   and  conversely. 


That  is, 
Proof 


{a/by  =  ay  b". 

f«Y^^l.«.«...tonfactors 
\bj       b    b    b 


Ex.  1. 


Ex.  2. 


(- 


_  aaa  •  -  •  to  n  factors 
~  bbb  ••' to  n  factors 
=  a'yb\ 


27  a^^c^ 


by  notation 

§91 

by  notation 
§§  119,  186 
§§  118,  119 


(x'^-7x+  12)2 
{X  -  3)-^ 


^IM"^^^-^^-^'-    5 


x-S 
Exercise  75 


186 


Write   each   of  the  following  powers  as  a  quotient  of 
products : 

2a\2  ^     r_2_ax\^  ^     /2a^Y 

SbfJ'  '    \3bYj' 


1. 


'zay 

3  by' 

f     3  ax'y 


3. 


V     2  62/7  '  V     2  6cVy  * 


FRACTIONS  186 

'■(-?)■  -MS)'  "-(-sr 

Simplify  each  of  the  following  expressions : 

(»-a/  *     (a; +  2/)"  '    (a' -by 

20.        (^^-^)*    .  22.    ^^-+iYx'^^^. 

(a^  +  a  +  iy  V2/  +  V      ^'  +  1 

\2ab'     mhjy2ab'     mh) 
Expand  each  of  the  following  powers : 

-g.!)-.  »-($-^-  ...e.^jj. 

-i^'W  -■($-?)■  -(^^o■ 

V6     .yy  \h     a)  \x     a     yj 

Factor  each  of  the  following  expressions : 

34.  i-^+i-  '       36.  %-"^+i- 

y     y-  y      by      b^ 

35.   i!i4.1^'4.25.  37.   -i^-^4-4. 

y        y  4/      y 


186  ELEMENTS   OF  ALGEBRA 

38.  51^  +  52£  +  4. 

39.  •^-2+-^. 
25  9^ 

41.  -^  +  1^  +  40^  +  8. 

27        3 

7/^      yiQ     9  2/^      x^     2ax     o? 

42.  8ar^-4;.y  +  |V-A.    48.    _--4.___. 

*    642/3      8/^         ■  '    25a2"^5a2'^a2       c^  * 

Eeduce  to  its  lowest  terms  each  of  the  following  fractions : 


44. 

45. 

4aV      9?/' 
6^          c^ 

46. 

47. 

40)2      4_^4aj_ 

a" 

51. 


a;2_  30^2/ -28^2 


64. 

a.'«  + 

x'- 

x'-l 

a.-«- 

x«  + 

x'-l 

55. 

a^- 

-a^6 

-  ab*  +  b' 

a'- 

a3/>- 

-  a'b'  + 

a63 

52     (^'  -  a^)  (g  4-  a;)  gg     (x-^y  ^zf-jx-y -zf 

(a^-^x'){a-x)  '         3x(y''-\-2yz-\-z^) 

gg      (x'-f)(x-y)  g^     a'-16 

{^-f){x^-y^)  '   a*-4a3+8a2_l6a+16 


CHAPTER   XIII 
FRACTIONAL  EQUATIONS 

187.  A  fractional  equation  is  an  equation  one  or  both  of 
whose  members  are  fractional  with  respect  to  an  unknown. 

E.g., =  4  is  a  fractional  equation  in  x, 

2a;  -  1         X 

while  ^  -I-  ^  =  ^sd: —  is  an  integral  equation  in  x. 

We  cannot  speak  of  the  degree  of  a  fractional  equation. 
The  term  degree  as  defined  in  §  101  applies  only  to  an 
integral  equation. 

188.  If  both  members  of  an  integral  equation  are  mxdti plied 
by  the  same  unknown  integral  expression  M,  the  derived  equa- 
tion has  all  the  roots  of  the  given  equation,  and,  in  addition, 
those  of  M=0. 

E.g..,  if  we  multiply  both  members  of  the  equation 

2x+l=a;  +  3  (1) 

by  ic  —  5  ;  the  root  5  is  introduced  in  the  derived  equation. 

Proof  If  A  and  B  denote  integral  expressions  in  the 
unknown,  and  we  multiply  both  members  of 

A  =  B  (1) 

by  any  unknown  integral  expression  M,  we  obtain 

AM=  BM,  or  {A  -  B)  M=  0.  (2) 

By  §  149,  (2)  is  equivalent  to  the  two  equations 

^-^  =  0and3/=0. 
187 


188  ELEMENTS  OF  ALGEBRA 

That  is,  the  roots  of  M=0  are  introduced  in  the  derived 
equation  (2)  by  multiplying  both  members  of  (1)  by  M. 

189.  If  both  members  of  a  fractional  equation  in  one  un- 
known are  multiplied  by  any  integral  expression  which  is 
necessary  to  clear  the  equation  of  fractions,  the  derived  integral 
equation  will  be  equivalent  to  the  given  fractional  equation. 

3 

Ex.  1.    Solve  the  equation  =  5  —  x.  (1) 

X  —\ 

Multiplying  by  ic  —  1  to  clear  (1)  of  fractions,  we  obtain 

3  =  6  X  -  x2  -  5. 

Transpose,  x2  -  6  x  +  8  =  0. 

Factor,  (x  -  2)  (x  -  4)  =  0.  (2) 

No  root  could  be  lost,  nor  could  either  root  of  (2)  be  introduced  by 
multiplying  (1)  by  x  —  1 ;  hence  (2)  is  equivalent  to  (1). 
Therefore,  the  roots  of  (1)  are  2  and  4. 

Ex.  2.    Solve  -^  +  -^  =  5.  (1) 

X  —  5     X  —  3 

Multiplying  by  (x  —  5)  (x  —  3)  to  clear  (1)  of  fractions  we  obtain 

3  (X  -  3)  +  2  X  (x  -  5)  =  5  (X  -  5)  (x  -  3). 

.  •.  x2  -  11  X  +  28  =^  0. 

...  (x_4)(x-7)=0.  (2) 

No  root  could  be  lost  nor  could  either  root  of  (2)  be  introduced  by 
multiplying  by  x  —  5  or  x  —  3  ;  hence  (2)  is  equivalent  to  (1). 
Therefore,  the  roots  of  (1)  are  4  and  7. 

Proof  By  transposing  to  the  first  member  all  the  terms 
of  any  fractional  equation,  adding  them,  and  reducing  the 
resulting  fraction  to  its  lowest  terms,  we  derive  an  equation 
of  the  form 

A/B  =  0,  (1) 

where  J.  and  jB  have  no  common  factors. 


FRACTIONAL  EQUATIONS  189 

By  the  preceding  principles  of  equivalent  equations,  the 
derived  equation  (1)  is  equivalent  to  the  given  fractional 
equation. 

We  are  to  prove  that  (1)  is  equivalent  to  the  equation 

^  =  0.  (2) 

Any  root  of  (1)  reduces  A/B  to  0.     But  when  A/B  is 
zero,  A  is  zero ;  hence  any  root  of  (1)  is  a  root  of  (2). 
To  prove  the  converse  we  must  first  prove  that  when 

A  =  0,  B=^0. 

If  A  and  B  could  become  0  for  the  same  value  of  x,  as  a ; 
then  by  §  132  they  would  have  the  factor  x  —  a  in  common. 
But  by  hypothesis  A  and  B  have  no  common  factor ;  hence 
when  A  =  0,  B^O. 

Hence  any  root  of  (2)  reduces  ^  to  0  but  not  B  to  0. 

Therefore  any  root  of  (2)  reduces  A/B  to  0  and  is  a  root 

of  (1). 

Hence  equations  (1)  and  (2)  are  equivalent. 

Ex.  3.    Solve  1  -  -^  =  — 6.  (1) 

Transposing  and  adding  the  fractions,  we  have 

r'2  —  1 
1  - +  0  =  0. 

X  —  1 

.  •.  1  -  (x  +  1)  +  G  =  0,  or  a;  =  6.  (2) 

By  §§  105  and  100,  equation  (2)  is  equivalent  to  (1)  ;  hence  0  is 
the  one  and  only  root  of  (!)• 

But  if,  as  would  be  more  natural  for  the  beginner,  we  should 
clear  equation  (1)  of  fractions  by  multiplying  by  x  —  1,  we  would 
obtain 

x-l-x2  =  -l-Gx  +  e. 

Transpose,  x^  -  7  x  +  0  =  0. 

...(x-i)(x-e)=0,  (3) 

of  which  the  roots  are  1  and  0. 


100  ELEMENTS   OF  ALGEBRA 

As  was  shown  above,  to  clear  equation  (1)  of  fractions  it  was  not 
necessary  to  multiply  by  x  —  \  ;  hence  multiplying  (1)  by  x  —  1  is 
the  same  as  multiplying  the  equivalent  integral  equation  (2)  by  ic  —  1. 

In  clearing  of  fractions  an  equation  in  one  unknown,  to 
avoid  introducing  roots,  the  following  suggestions  should 
be  heeded : 

(i)  Fractions  having  a  common  denominator  should  be 
combined. 

(ii)  Factors  common  to  the  numerator  and  denominator 
of  any  fraction  should  be  cancelled. 

(iii)  When  multiplying  by  a  multiple  of  the  denomi- 
nators, we  should  always  use  the  L.  C.  M. 

Ex.  4.    Solve         -^—  +  ^^  =  ^i-i  +  ^^=-^.  (1) 

x  —  'l      X  —  1      x—\      X  —  iS 

Transpose  so  that  each  member  is  a  difference, 

x         X  4-  1  _x  —  8      a;  —  9 
05  —  2      X  —  \      X— 6      X  —  7 

Combine,  ^ = ? :  (2) 

(x  _  2)  (X  -  1)      (x  -  7)  (X  -  6)  '  ^  ^ 

Clear  of  fractions,  x2  -  1.3  x  +  42  =  x2  -  3  x  +  2. 

.  •.  10  X  =  40,  or  X  =  4.  (3) 

Since  the  root  4  could  not  be  introduced  in  clearing  (2)  of  fractions, 
4  is  the  root  of  (1). 

Ex.  5.    Solve         x-^_^x±^^x±\  ^x  +  3,  ^^ 

x+lx+7x+3x+5 

Transpose,  x-^  _x±l^x±^  _x±b^  ^^ 

x+lx+3x+5x+7 

Combine,  ^- = ^ (3) 

(X  +  1)  (X  +  3)      (X  +  5)  (X  +  7) 

Clear  of  fractions,.x2  +  12x  +35=  x^  -f  4  x  +  3. 

.•.8x  =  -32,  or  x  =  -4. 


FRACTIONAL   EQUATIONS  191 

Or  reducing  the  improper  fractions  in  (2)  to  mixed  expressions,  we 
have, 


x+ I  x+3  x+5  x+7 

1       .       I  ^      +      ^ 


x-\-l      x  +  3         x  +  5      x  +  7 

Combining  these  fractions,  we  obtain  equation  (3)  above. 
Since  the   root   —  4  could  not  be  introduced  in  clearing  (3)  of 
fractions,  —  4  is  the  root  of  (I). 


Exercise  76. 
Solve  each  of  the  following  fractional  equations : 
,     3x-16     5 


X  3 

5x-5 


11. 


x-\-l 

a;-1^2 
x  +  1     3 

x-2  ^ 
2x-5 


=  3.  12. 


X- 

-5 

2x- 

-2 

4a;- 

-5 

13. 


14. 


5.  2£:z3  =  .^_,.  15. 
3a;-4      6a;-7 

6.  -^—  =  -^-2.  16. 
x-\-l      x  -\-2 

7.  ^^-1-1  =  1.  17. 
X  -\-l      X 

8.  10^4^  =  ._A__7.  18.    ^ 


a  —  1        »  +  1 
1       .       1 


4a;+6     6a;+4     2£c+3 


19. 


12  2 

10.    — 1 = —  =  -^ —      20. 

3a;-f-9      5a;H-l      x+3  Sx-5     4a;  +  8 


^    1 

5              7 

2a;+3     4a?-f6     6a;+8 

4.^ 

X     _3 

a;  4-1 

a;-2-^' 

6x 

^     -5 

x-7 

x-6     ^• 

2x 

4a;        ^ 

x  +  3 

x  +  7      - 

1     + 
a;  +  4 

2            3 

a;-f-6     a;-h5 

3 

2            1 

x  +  1 

a;  +  2      a;H-3 

5     1 
2x  +  4: 

_3     1      ,      1 

3     1 

2xH-2 

=5     1     +1. 
4a;  +  3     4a; 

2x-5 

2a;-7 

3a;-7 

3a;-5 

6a;-2 

_3a;  +  7 

192                         ELEMENTS  OF  ALGEBRA 

21.  ^±l_^Z:i?  =  §.  25.    3'^-^-\-2^±l  =  5. 
ic  —  lie  +  3a;  ic  +  1        x  —  1 

22.  ^+2_£^  =  _§_.  26.   5^!^^-2^^  =  3. 
x-2     x  +  2     x  +  1  x  +  2       x  +  S 


x-\-3     ic  —  4       ic  x^  —  1     aj  +  l      1  —  ic 

24.    -^  =  3^^ ^.     28.    -^+      1  ^ 


a;-h2        i«-2     ic  +  l  x--9     x  +  3     3 


29. 


30. 


35. 
36. 
37. 
38. 
39. 


3  g;  -f  5  5       ^  8  4-  3  a; 

3a;-l     l--9a;*-^~l  +  3;»* 

1111 


a; +  5     x  +  6     x  +  6     aj  +  8 


31.       1     +_!=!     +     1 


x  +  2     x  +  10     a;  +  4     a  +  S 

32.    -J_  +  _1_  =  ^_  +  _L.. 

if  —  5     37  +  2     .T  —  4     x  +  1 

33         ^'      .a;  —  9_a;  +  l  ,  x  —  S 


x—2     x—1     x—1     X— 6 
g^     a;4-3      a;  — 6_a;  +  4  .  a;  — 5 


a;  +  l      a;  — 4      a;  +  2     x  —  3 

x  —  3      a;  — 4_a^  — 6      x  —  7 
a;  — 4     x  —  b     x  —  1     ^  —  8 

X         a;  +  l_a;  —  8      a;  — 9 
x  —  2      a;— l~a;  — 6      x  —  1 

x-[-iS  _  x  —  Q  _  x  —  4:  _  a;  —  15 
a;  +  4     x  —  1     x  —  5     x  — 16 

x—1  _  x  —  9  _ x  — 13  _  x  — 15_ 
X  — 9     a;  — ll~x— 15      x  —  ll' 

x-\-3 _ x  +  6  ^ x-\-2 _  a;4-5_ 
x4-6     x  +  d~x-{-5     x-i-S 


FJi ACTIONAL  EQUATIONS  193 


a? +  2      x  —  7     x-^3  __x  —  0 
X         x  —  5      x-\-l      if  —  4 

4a;_17     10a;_13     8aj-30,5a; 


a;_4  2a;-3        2a;-7        x-l 

^„     5a;-8  ,  6x-44     10a;-8     x-S 
x-2         x-7.        x-1        x-6 

^^    S0±6x_^60±8x^^^_^    48 


cc-f-l  x-i-3  ic-fl 

44     25-^a;     16a^  +  4i^g        23 
a;  +  l  3a;  +  2  aj  +  l 

45.         3_4-...^'        =;^+       ' 


46. 


4-2a;     8(l-x)      2  -  a;     2-2a; 
60      _     10^ 8_^ 


X 


4     5aj-30     3a;-12 


47.  -i 2-  =  ^ ?i-. 

a!  +  3     K  +  l     2x  +  6     2x  +  2 

48.  (2x-l)(3x  +  8)     j^Q_ 

6a;(a;  +  4) 

In  the  five  following  examples  first  reduce  improper  fractions  to 
mixed  expressions. 

4^     5ar-64     2  ic  - 11      4a;-55     aj~6 


50. 


51. 


52. 


a; -13 

X- 

-6 

a;- 

-14 

a; 

-7 

x  —  H    , 

.T-4 

a; 

-^- 

a:-7 
a;-9 

a;- 10  ' 

a;-G 

x-\-b  _^x 

•  +  c 

2. 

a;—  c 


a;  —  6     x-\-  c  x 


vnx  nx 

53.    ; 1 =  m  4-  w. 

m  4-  a;     n  -\-  x 


194  ELEMENTS  OF  ALGEBRA 

54. 


55. 


x  —  c_  x  —  b      2(b  —  c) 
x—b      X  —  c     X  —  b  —  c 

m  -f-  r         n-\-  r    _m  -\-  n  -\-  2  r 
x-\-2  n      x-\-2m       x-\-m-\-n 


190.   Problems  which  lead  to  fractional  equations. 

Prob.  1.    The  quotient  of  a  certain  number  increased  by  7  divided 
by  the  same  number  diminished  by  5  is  4.     Find  the  number. 
Let  X  =  the  required  number. 
Then  by  the  conditions  of  the  problem,  we  have 

^  +  7^1 
x  —  b 

Whence  a;  =  9,  the  required  number. 

Prob.  2.  The  value  of  a  fraction  is  1/4.  If  its  numerator  is  dimin- 
ished by  2  and  its  denominator  is  increased  by  2,  the  resulting  fraction 
will  be  equal  to  1/9.    Find  the  fraction. 

Let  X  =■  the  numerator  of  the  fraction  ; 

then  4  X  =  the  denominator  of  the  fraction  ; 

and,  by  the  conditions  of  the  problem,  we  have 

x-2  ^1 

4  a;  +  2      9' 

Whence  a;  =  4,  and  the  required  fraction  is  4/16. 

Exercise  77. 

1.  The  value  of  a  fraction  is  1/7.  If  its  numerator  is 
increased  by  5  and  its  denominator  by  15,  the  resulting 
fraction  will  be  equal  to  1/5.     Find  the  fraction. 

2.  The  sum  of  two  numbers  is  20,  and  the  quotient  of 
the  less  divided  by  the  greater  is  1/3.     Find  the  numbers. 

3.  What  number  added  to  the  numerator  and  denominator 
of  the  fraction  3/7  will  give  a  fraction  equal  to  2/3  ? 

4.  What  number  must  be  added  to  the  numerator  and 
subtracted  from  the  denominator  of  the  fraction  5/11,  to 
give  its  reciprocal? 


FRACTIONAL   EQUATIONS  195 

5.  The  reciprocal  of  a  number  is  equal  to  7  times  the 
reciprocal  of  the  sum  of  the  number  and  5.    Find  the  number. 

6.  A  train  ran  240  miles  in  a  certain  time.  If  it  had 
run  6  miles  an  hour  faster,  it  would  have  run  48  miles 
farther  in  the  same  time.     Find  the  rate  of  the  train. 

7.  A  number  has  three  digits  which  increase  by  2  from 
right  to  left.  The  quotient  of  the  number  divided  by  the 
sum  of  the  digits  is  48.     Find  the  number. 

8.  A  steamer  can  run  18  miles  an  hour  in  still  water. 
If  it  can  run  96  miles  with  the  current  in  the  same  time 
that  it  can  run  48  miles  against  the  current,  what  is  the 
rate  of  the  current  ? 

9.  A  number  of  men  have  $  80  to  divide.  If  $  150  were 
divided  among  2  more  men,  each  one  would  receive  $  5  more. 
Find  the  number  of  men. 

10.  The  circumference  of  the  hind  wheel  of  a  wagon 
exceeds  the  circumference  of  the  front  wheel  by  4  feet.  In 
running  200  yards  the  front  wheel  makes  10  more  revolu- 
tions than  the  hind  wheel.  What  is  the  circumference  of 
each  wheel  ? 

11.  A  number  has  two  digits  which  increase  by  4  from 
right  to  left.  If  the  digits  are  interchanged  and  the  result- 
ing number  is  divided  by  the  first  number  the  quotient  will 
be  4/7.     Find  the  number. 

12.  A  train  runs  10  miles  farther  in  an  hour  than  a  man 
rides  on  a  bicycle  in  the  same  time.  If  it  takes  the  man  5 
hours  longer  to  ride  352  miles  than  it  takes  the  train  to  run 
the  same  distance,  what  is  the  rate  of  the  train  ? 

13.  A  tank  can  be  filled  with  one  pipe  in  30  minutes,  by 
a  second  pipe  in  40  minutes,  by  a  third  in  50  minutes.  How 
long  will  it  take  to  nil  it  with  them  all  running  together  ? 

14.  A  can  do  a  piece  of  work  in  3^  days,  B  in  2i  days, 
C  in  3|  days.  If  A,  B,  and  C  work  together,  how  long  will 
it  take  to  do  the  work  ? 


196  ELEMENTS   OF  ALGEBRA 

15.  A  cistern  can  be  filled  in  15  minutes  by  two  pipes,  A 
and  B,  running  together ;  after  A  has  been  running  by  itself 
for  5  minutes,  B  is  also  turned  on,  and  the  cistern  is  filled 
in  13  minutes  more.  In  what  time  would  it  be  filled  by 
each  pipe  separately  ? 

16.  A  man,  woman,  and  child  could  reap  a  field  in  30 
hours,  the  man  doing  half  as  much  again  as  the  woman,  and 
the  woman  two-thirds  as  much  again  as  the  child.  How 
many,  hours  would  they  each  take  to  do  it  separately  ? 

17.  A  and  B  ride  100  miles  from  P  to  Q.  They  ride 
together  at  a  uniform  rate  until  they  are  within  30  miles  of 
Q,  when  A  increases  his  rate  by  1/5  of  his  previous  rate. 
When  B  is  within  20  miles  of  Q  he  increases  his  rate  by 
1/2  of  his  previous  rate,  and  arrives  at  Q  10  minutes  earlier 
than  A.     At  what  rate  did  A  and  B  first  ride  ? 

18.  A  and  B  can  reap  a  field  together  in  12  hours,  A  and 
C  in  16  hours,  and  A  by  himself  in  20  hours.  In  what 
time  could  B  and  C  together  reap  it  ?  In  what  time  could 
A,  B,  and  C  together  reap  it  ? 

19.  The  sum  of  two  numbers  is  n,  and  the  quotient  of  the 
less  divided  by  the  greater  is  a/b.     Find  the  numbers. 

20.  The  reciprocal  of  a  number  is  n  times  the  reciprocal 
of  the  sum  of  the  number  and  a.     Find  the  number. 

21.  A  train  ran  a  miles  in  a  certain  time.  If  it  had  run 
b  miles  an  hour  faster,  it  would  have  run  c  miles  further  in 
the  same  time.     Find  the  rate  of  the  train. 

22.  A  steamer  can  run  a  miles  an  hour  in  still  water.  If 
it  can  run  b  miles  with  the  current  in  the  same  time  that  it 
can  run  c  miles  against  the  current,  what  is  the  rate  of  the 
current  ? 

23.  The  value  of  a  fraction  is  1/a.  If  its  numerator  is 
increased  by  m  and  its  denominator  by  n,  the  resulting 
fraction  will  be  equal  to  1/6.     Find  the  fraction. 


CHAPTER  XIV 
SYSTEMS   OF  LINEAR   EQUATIONS 

191.  Equations  in  two  or  more  unknowns.    In  the  equation 

y  =  3x  +  2,  (1) 

where  x  and  y  are  both  unknowns,  y  has  one  and  only  one 
value  for  each  value  of  x. 

E.g.,  when  x  =  l,  y  =  5]  when  x  =  2,  y  =  S  -,  when  x  =  3, 
?/  =  11 ;  when  a;  =  4,  y  =  14,  etc. 

That  is,  equation  (1)  restricts  x  and  y  to  sets  of  values. 

In  like  manner,  any  equation  in  two  or  more  unknowns 
restricts  its  unknowns  to  sets  of  values. 

192.  A  solution  of  an  equation  in  two  or  more  unknowns 
is  any  set  of  values  of  the  unknowns  which  renders  the 
equation  an  identity. 

E.g.,  if  in  the  equation 

2/  =  3a; +  2  (1) 

we  put  3  for  x  aud  11  for  y,  we  obtain  the  identity 

11  =  3  X  3  +  2. 

Hence  3  and  11,  as  a  set  of  values  of  x  and  y,  is  one  solution  of  (1)  ; 
2  and  8  is  another  solution  ;  and  so  on. 

Note.  The  word  solution  denotes  either  the  process  of  solving  or 
the  result  obtained  by  solving.  The  word  is  here  used  in  the  latter 
sense. 

A  root  of  an  equation  in  one  unknown  is  often  called  a  solution. 

197 


198  ELEMENTS   OF  ALGEBRA 

193.  The  degree  of  an  integral  equation  in  two  or  more 
unknowns  is  the  degree  of  that  term  which  is  of  the  highest 
degree  in  the  unknowns. 

E.g.,  ax  -^  by  =  7  is  R  linear  equation  in  x  and  y  ;  while  ax^  -\-hy  =  c 
or  cxy  +  3  ic  =  2  is  a  quadratic  equation  in  x  and  y. 

194.  Two  equations  are  said  to  be  equivalent  when  every 
solution  of  each  equation  is  a  solution  of  the  other. 

195.  The  following  principles  concerning  the  equivalence  of 
equations,  which  have  been  proved  for  equations  in  one 
unknown,  hold  true  for  all  equations : 

(i)  If  for  any  exj^ression  in  an  equation  an  identical  expres- 
sion is  substituted,  the  derived  equation  will  he  equivalent  to 
the  given  one  (^  105). 

(ii)  If  identical  expressioiis  are  added  to  or  subtracted  from 
both  members  of  an  equation,  the  derived  equation  will  be 
equivalent  to  the  given  one  (§  106). 

(iii)  If  both  members  of  an  equation  are  midtiplied  or 
divided  by  the  same  known  expression,  not  denoting  zero,  the 
derived  equation  ivill  be  equivalent  to  the  given  one 
(§§  108,  110). 

(iv)  If  one  member  of  an  equation  is  zero,  and  the  other 
member  is  the  jyroduct  of  two  or  more  integral  factors,  the 
equations  formed  by  putting  each  of  these  factors  equal  to  zero 
are  together  equivalent  to  the  given  equation  (§  149). 

E.g.,  the  equation 

(ic  +  2?/-4)(2a;-32/+  1)  =  0 
is  equivalent  to  the  two  equations 

x  +  2i/-4  =  0  and  2a;-32^  +  l=0. 

(v)  If  both  members  of  an  integral  equation  are  multiplied 
by  the  same  unknown  integral  exjyressioyi  M,  the  derived  equa- 
tion has  all  the  solutions  of  the  given  equation,  and  in  addition 
those  of  M=0  (§188). 


SYSTEMS  OF  LINEAR   EQUATIONS  199 

Proof.  If  in  the  proofs  of  these  principles  for  equations  in  one 
unknown  we  substitute  the  word  "solution"  for  the  word  "root," 
the  proofs  will  apply  to  equations  in  any  number  of  unknowns. 

Exercise  78. 

Of  the  following  equations  state  which  are  equivalent  to 
the  equation  2  a;  +  ?/  =  3,  and  give  the  reason  : 

1.  (4  a- -f  2  2/)/2  =  3.  4.   6  a; +  3^  =  9. 

2.  'Sx-\-y  =  x  +  ^.  5.    (2a;4-?/)/3  =  l. 

3.  a;  +  y  =  3  —  x.  Q.   i:X  +  oy  =  (S-\-y. 

State  to  what  two  linear  equations  each  of  the  following 
quadratic  equations  is  equivalent,  and  give  the  reason : 

7.    (a;-2/)(a-+2y-hl)=0.      8.    {■y-x)x-^2y{x-y)=0. 

Obtain  ten  solutions  of  each  of  the  following  equations : 
9.    2a;-hy  =  3.      10.    2a; -f  3?/ =  6.      11.    2  a;  — 3y  =  4. 

12.  How  many  solutions  has  a  single  equation  in  two 
unknowns  ? 

13.  By  (iii)  in  §  195,  show  that  the  two  equations 

ax  -\-by  =  c  and  a'x  -\-  b'y  =  c' 
are  equivalent  when  a' /a  =  b'/b  =  c'/c. 

196.   Independent  equations. 

Prob.  If  the  sum  of  two  numbers  is  10  and  their  difference  is  4, 
what  are  the  two  numbers  ? 

Let  X  =  the  less  number 

and  y  =  the  greater  number. 

Then  by  the  Jirst  condition  we  have  the  equation 

y  +  x  =  \0;  (1) 

and  by  the  second  condition  wc  have  the  equation 

y  -  X  =  4.  (2) 


200  ELEMENTS  OF  ALGEBRA 

In  (1),  when  x  =  l,  y  ^9;  when  x  =  2,  y  =  S ;  when  x  =  3, 
/  =  7,  etc. 

In  (2),  when  x  =  1,  y  =  5 ;  when  x  =  2,  ?/  =  6;^when  jr  =  3, 
/  =  7,  etc. 

Hence,  3  and  7  is  a  set  of  values  of  x  and  y  which  will  satisfy  each 
of  the  two  different  conditions  expressed  by  equations  (1)  and  (2), 
and  are  therefore  the  required  numbers. 

Equations,  like  (1)  and  (2),  which  express  different  condi- 
tions are  called  independent  equations. 

Observe  that  independent  equations  express  different  rela- 
tions between  their  unknowns,  while  equivalent  equations 
express  the  same  relation. 

Any  solution  as  x  =  3,  ?/  =  7,  can  be  written  briefly  3,  7, 
it  being  understood  that  the  value  of  x  is  written  first. 

197.  Systems  of  equations.  Two  or  more  equations  are 
said  to  be  simultaneous,  when  the  unknowns  are  restricted 
to  the  set  or  sets  of  values  which  satisfy  all  the  equations. 

A  group  of  two  or  more  simultaneous  equations  is  called 
a  system  of  equations. 

E.g.^  equations  (1)  and  (2)  in  §  196  are  simultaneous^  and  form  a 
system  of  equations. 

198.  A  solution  of  a  system  of  equations  is  a  set  of  values 
of  its  unknowns  which  satisfies  all  its  equations. 

E.g.,  3,  7  is  a  solution  of  the  system  of  equations,  (1) 
and  (2),  in  §  196. 

To  solve  a  system  of  equations  is  to  find  all  its  solutions. 

199.  Equivalent  systems.  Two  systems  of  equations  are 
said  to  be  equivalent  when  every  solution  of  each  system  is 
a  solution  of  the  other  system. 


E.g.^  the  systems  (a)  and  (6) 

x^2y  =  b,         (1)1 
4.x-    y  =  2,        (2)}*^'^'        i(5-2y)-y  =  2,       (4)  j 


x-^2y  =  b,         (1)1       ,  x  =  5-2y,  (3)1^^^ 

[  («)      ...    ^  .         ^      ...  r  (ft) 


SYSTEMS  OF  LINEAR  EQUATIONS  201 

are  equivalent ;  for  each  system  has  the  solution  1,  2  ;  and,  as  will  be 
proved  later,  neither  system  has  any  other  solution. 

Observe  that  (3)  is  obtained  by  solving  (1)  for  x,  and  (4)  by  put- 
ting in  (2)  the  value  of  x  given  in  (3) ;  x  therefore  does  not  appear  in 
(4),  and  is  said  to  have  been  eliminated. 

200.  Elimination  is  the  process  of  deriving  from  two  or 
more  equations  a  new  equation  involving  one  less  unknown 
than  the  equations  from  which  it  is  derived. 

The  unknown  which  does  not  appear  in  the  derived  equa- 
tion is  said  to  have  been  eliminated;  as  a;  in  §  199. 
There  are  in  common  use  two  methods  of  elimination : 

I.  Elimination  by  substitution  or  comparison. 

II.  Elimination  by  addition  or  subtraction. 

201.  In  this  chapter  we  shall  use  three  principles  con- 
cerning the  equivalence  of  systems  of  equations.  For  con- 
venience of  reference  we  shall  number  them,  (i),  (ii),  (iii). 

(i)  Equivalent  equations.  If  any  equcUion  of  a  system  is 
replaced  by  an  equivalent  equation,  the  derived  system  will  be 
equivalent  to  the  given  system. 

E.g.,  since  equation  (3)  is  equivalent  to  (1),  and  (4)  to  (2),  system 
(6)  is  equivalent  to  system  (a). 


3a;  +  22/  =  8,         (1)1                    dx 
4x-Sy  =  5,        (2)/^^^             8x 

+  6y  = 
-Gy  = 

=  24, 
=  10. 

>> 

The  only  solution  of  either  system  is  2,  1. 

Ex.   Solve  the  system        3  +  4  a;  =  15, 

Sl<-' 

2  +  3y  =  8. 

From  (1),                                    a;  =  3. 
From  (2),                                    y  =  2. 

S)<'> 

Since  equation  (3)  is  equivalent  to  (1),  and  (4)  to  (2),  system  (b) 
is  equivalent  to  system  (a) ;  hence,  the  one  and  only  solution  of  system 
(a)  is  3,  2. 


202  ELEMENTS   OF  ALGEBBA 

Proof  of  (i).  Let  (1)  and  (2)  be  a  system  of  equations  in 
two  unknowns, 

^  =  ^'  ^^A(a^  "^'  =  ^''  ^^^(h^ 

C=D,  (2)r^  C'  =  Z>'  (4)]^^ 

and  let  (3)  be  equivalent  to  (1),  and  (4)  to  (2) ;  we  are  to 
prove  that  system  (h)  is  equivalent  to  system  {a). 

Since  (1)  and  (3)  have  the  same  solutions,  and  (2)  and  (4) 
also  have  the  same  solutions ;  it  follows  that  any  solution 
common  to  (1)  and  (2)  will  be  common  to  (3)  and  (4)  also ; 
and  conversely.     Hence,  systems  (a)  and  (6)  are  equivalent. 

202.  The  method  of  elimination  by  substitution  depends 
upon  the  following  principle  of  equivalence  of  systems: 

(ii)  Substitution.  If  one  equation  of  a  system  is  solved  for 
one  of  its  unknowns,  and  the  value  thus  obtained  is  substituted 
for  this  unknown  in  the  other  equation  (or  equations)  of  the 
system,  the  derived  system  will  be  equivalent  to  the  given  one. 


Ex.  1.    Solve  the  system  2  a:;  =  10,  (1) 

y  =  l2-lSx.  (2)  j 


■(«) 


From  (1),  X  =  5.  1 

Substituting  this  value  of  x  in  (2),  we  obtain  I  (?)) 

ij  =  12 -16  =  -3.  J 

By  (ii),  system  (6)  is  equivalent  to  system  (a). 

Hence,  the  one  and  only  solution  of  system  (6)  or  (a)  is  5,  —  3. 

Ex.  2.    Solve  the  system     Sx  +  by  =  19,  (1)  1 

bx-4y  =  1.  (2)  J 

rrom(l),  x  =  il9-by)/S.  (3)1 

Substituting  this  value  for  x  in  (2),  we  have  [  (6) 

(5/3)(19-5i/)-42/  =  7.  (4)  J 


SYSTEMS   OF  LINEAR  EQUATIONS  203 


By  (ii),  system  (6)  is  equivalent  to  system  (a). 

From  (4),  y  =  2.  (5) 

Substituting  2  for  y  in  (3),  we  obtain 

x=(19-10)/3  =  3.  (6) 


(c) 


By  (ii),  system  (c)  is  equivalent  to  system  (6)  ;  hence,  the  one  and 
only  solution  of  system  (c)  or  its  equivalent  system  (a)  is  3,  2. 


Ex.  3.   Solve  the  system    2x-  5y=l,  (1) 

7x  +  3i/  =  24.  (2)' 


}(«: 


From(l),  x=(5y+l)/2.  (3)1 

Substituting  this  value  of  x  in  (2),  we  have  >  (/>) 

(7/2)(5y+l)+3y  =  24.  (4)  J 

By  (ii),  system  (6)  is  equivalent  to  system  (a). 

From  (4),  y  =  1.  (5) 

Substituting  this  value  of  y  in  (3),  we  obtain  (c) 

x  =  (5  +  l)/2  =  3.  (6). 

By  (ii),  system  (c)  is  equivalent  to  (6),  and  therefore  to  (a). 
Hence,  the  one  and  only  solution  of  system  (a)  is  3,  1. 

The  foregoing  examples  illustrate  the  following  rule  for 
eliminating  by  substitution. 

From  one  of  the  equations  find  the  value  of  the  unknown  to 
he  eliminated,  in  terms  of  the  others;  then  substitute  this  value 
for  that  unknown  in  the  other  equation  or  equations. 

Proof  of  (ii).  Let  (1)  and  (2)  be  a  system  in  two 
unknowns,  and  let  (3)  be  the  equation  obtained  by  solving 


^  =  A  (1)K.  =^  =  P,  (3)1 

C  =  D,  {2)]^'  C'  =  D',  (4)J 


(b) 


(1)  for  X,  and  (4)  the  equation  obtained  by  substituting  for  x 
in  (2)  its  value  F  as  given  in  (3);  we  are  to  prove  that 
system  (6)  is  equivalent  to  system  (a). 


204 


ELEMENTS  OF  ALGEBRA 


Since  x  =  Fis  equivalent  to  equation  (1),  system  (c)  is  by 
(i)  equivalent  to  system  (a), 

X=:F,  (5)1 


C  =  D. 


(6)  J 


(«) 


Any  solution  of  system  (c)  renders  x  =  F  and  C=D', 
hence,  any  such  solution  must  satisfy  (6)  after  F  has  been 
substituted  for  x  (vi,  §  32) ;  therefore,  any  solution  of 
system  (c)  will  be  a  solution  of  system  (b). 

Conversely,  any  solution  of  system  (b)  renders  x  =  F  and 
C  =  D' ;  hence,  any  such  solution  must  satisfy  (4)  after  x 
has  been  substituted  for  F  (vi,  §  32) ;  hence,  any  solution  of 
system  (b)  is  a  solution  of  (c). 

Hence,  system  (b)  is  equivalent  to  system  (c)  or  (a). 

In  like  manner  the  theorem  could  be  proved  if  the  systems 
(a)  and  (6)  contained  three  or  more  equations. 


Exercise  70. 

Solve  each  of  the  following  systems  of  equations  by  the 
method  of  substitution : 


1.  3  a;  =  27, 

2  a;  4-  3  2/  =  24. 

2.  3  a; +  4  2/ =  58,' 
2y  =  U, 

3.  3  a; +  4  2/ =  10,1 
4  a;  +  2/  =  9.       J 

4.  x-\-2y  =  lS, 
Sx  + 

5. 


2/  =  13,| 
2/  =  14.  J 
4x  +  72/  =  29,l 
a;  +  3  2/  =  11.     J 
6.    5  a; +  6  2/ =  17,1 
6  a;  +  5  2/  =  16.  J 


7.  8  a; -2/ =  34,1 
ic  +  8  2/  =  53. 1 

8.  6^ -5  a;  =  18,1 
12  a;  -  9  2/  =  0.  J 

9.  7aj  +  42/  =  l,  I 
9a;  +  4^  =  3.  J 

10.  x-lly  =  l, 
111  2/  -  9  a;  =  99. 

11.  3  a; +  5  2/ =  19,1 
5x  —  4:y  =  7.    J 

12.  8  x- 21 2/ =  5,       1 
6  a;  +  14  2/  =  -  26.  J 


SYSTEMS  OF  LINEAR  EQUATIONS  205 

13.  3  a; -112/ =  0,    I  16.    1 0^  +  3  2/ +  14  =  0,1 
19 a;- 19 2/ =  8.  J  |a;  +  52/  +  4  =  0.    1 

14.  ^^-^^y  =  ^'_  ]  17.    i(^+x)=i(9+y),         I 


25  0.-17  2/ =  139.  J  i(ll+^+2/)=i(9+2/) 

15.    ^_^_1 
3      6~2' 

5      10      2  J 


18.    J(a.  +  1)=4(2/  +  2),| 
i(^  +  2/)=i(2/  +  2).J 


19.    (a;  +  l)(2/  +  5)  =  (.T  +  5)(2/  +  l),| 


] 


20.    an/-(a;-l)(2/-l)=6(2/-l) 
a;  -  2/  =  1. 


203.  The  following  example  illustrates  a  special  form  of 
the  method  of  elimination  by  substitution,  which  is  called 
elimination  by  comparison. 


Ex.   Solve  the  system 

2x-3y  =  \, 
6x-^2y  =  l2Q. 

;:;)'•' 

Solve  (1)  for  a;, 
Solve  (2)  for  a;, 

a;  =  (3y  +  l)/2. 
x=026-2y)/5. 

'> 

Substituting  in  (4)  the  value  of  x  given  in  (3),  or,  what  amounts  to 
the  same  thing,  putting  these  two  values  of  x  equal  to  each  other,  we 
obtain 

3y4-1^126-y    ^r  y  =  13. 


2  6 

Substituting  in  (3),        x  =  40/2  =  20. 


(c) 


By  principles  (i)  and  (ii),  systems  (a)  and  (c)  are  equivalent ;  or, 
in  other  words,  no  solution  has  been  either  lost  or  introduced  in  pass- 
ing from  system  (a)  to  system  (c)  ;  hence,  the  one  and  only  solution 
of  system  (a)  is  20,  13. 


206 


ELEMENTS  OF  ALGEBRA 


Exercise  80. 

Solve  each  of  the  followmg  systems  by  the  method  of 
comparison : 


1.    ^+2/ =  16, 


4 


2.    X  —  y 


6.    ^  +  ^  =  5, 

5      2       • 

X  —  y  =z4. 


7.  l-^-TV?y  =  3,| 

4:X-y  =  20.     J 


-?  =  2. 


8.    ^  +  -^^ 


0, 


10, 


3.  ^  +  2/ 

1  i»  +  2/  =  50 

4.  a?  =  3  2/, 

1  oj  +  2/  =  34 
5  it* 


3      4 

Sx-7y  =  37. } 

X  -\-l  _3  y  —  5 

10    ~~^ 
a;  +  1      X  —  w 


5. 


2/ 


5  2/ 


10. 


10  8 

a^  +  3      8  -  V         I 

3  (x  +  ?/)  ^  a?  +  3 
8        ~     5    ' 


204.    The  method  of  elimination  by  addition  or  subtraction 

depends  upon  the  following  principle : 

(iii)  Addition.  If  an  equation  obtained  by  adding,  or  sub- 
tracting, the  corresponding  members  of  two  or  more  equation"} 
of  a  system  is  put  \n  the  place  of  any  one  of  these  equations, 
the  derived  system  will  be  equivalent  to  the  given  system. 

Ex.  1.   Solve  the  system    3  a;  +  7  y  =  27,  (1)  ) 

bx  +  2y  =  \Q.  (2)  J 

To  eliminate  x,  we  obtain  from  (1)  and  (2)  equivalent  equations  in 
which  the  coefficients  of  x  are  equal. 

Multiply  (1)  by  5,  15  x  +  35 «/  =  135,  (3) 


Multiply  (2)  by  3, 


Ibx  +  Qy 


48. 


(4) 


(&) 


SYSTEMS   OF  LINEAR   EQUATIONS  207 

Subtract  (4)  from  (3),  29  </  =  87,  (5) 

From  (4),  15  x  +  6  y  =  48.  (6) 

From  (5),  y  =  3, 

Substitute  in  (6),  x  =  2. 


lid) 


Proof  of  equivalency.  By  (i),  systems  (a)  and  (&)  are  equivalent; 
by  (iii),  system  (c)  is  equivalent  to  (b)  ;  and  by  (ii)  and  (i),  system 
(d)  is  equivalent  to  (c). 

Hence  the  one  and  only  solution  of  (a)  is  2,  3. 


Ex.2.  Solve  the  system     7  x  +  2  y  =  47,  ^_,  , 

ox  —  4  y  =  1. 


(2)1 


To  eliminate  y  we  obtain  from  (1)  and  (2)  equivalent  equations  in 
which  the  coefficients  of  y  are  arithmetically  equal. 

Multiply  (1)  by  2,  14  x  +  4  y  =  94.  (3)  ) 

From  (2),  6x-4?/=    1.  (4)  J 

Add  (3)  and  (4),     19  x  =  95,  or  x  =  5.  (5)  | 

Substitute  in  (4),  y  =  0.  j 

Proof  of  equivalency.  By  (i),  system  (h)  is  equivalent  to  (a)  ;  by 
(iii),  system  (4)  and  (5)  is  equivalent  to  (6)  ;  and  by  (ii)  and  (i), 
system  (c)  is  equivalent  to  system,  (4)  and  (5). 


Ex.  3.    Solve  the  system 

(X  -  l)(y  -  2)-(x  -  2)(y  -  1)  =  -  2,  (1) 

(X  +  2)(y  +  2)-(x  -  2)(y  -  2)=  32. .  (2) 

We  first  reduce  (1)  and  (2)  to  the  form  (ix+  by  =  c. 


|(«) 


From(l),  x-y  =  2,  ^^U  (6) 

From  (2),  x  +  y  =  8.  (4)  J 

Add  (3)  and  (4),  2  x  =  10,  or  x  =  5.  (5) 

Subtract  (3)  from  (4),  2  y  =  6,    or  y  =  3.  (6) 


By  (i),  system  (&)  is  equivalent  to  (a) ;  by  (!)  and  (iii),  either  (5) 
an^l  (4)  or  (6)  and  (4)  form  a  system  equivalent  to  (6)  ;  hence  the 
solution  of  (a)  is  given  in  (5)  and  (6). 


208  ELEMENTS   OF  ALGEBRA 

Ex.  4.   Solve  the  system 

Sx-l^  =  ^-^^,  (1) 

/  id 

We  first  reduce  (1)  and  (2)  to  the  form  ax  +  by  =  c. 


(a) 


From  (1), 
From  (2), 

Ux-2y  =  -Sl, 
10x  +  6y  =  S7. 

:,h 

Multiply  (3)  by  3, 

i2x-6y  =  -dS. 

(5) 

Add  (4)  and  (5), 
Substitute  in  (3), 

62  X  =  -  56,  or  ic  = 
y  =  207/26. 

-  14/13.      ) 

}(C) 

Since  (3)  and  (5)  are  equivalent  and  (3)  is  the  simpler  equation, 
to  find  y  we  substitute  in  (3)  rather  than  in  (5). 

By  (i),  (ii),  and  (iii),  no  solution  has  been  lost  or  introduced  in 
passing  from  system  (a)  to  system  (c). 

Proof  of  (iii).  Let  (1)  and  (2)  be  a  system  in  two 
unknowns, 

^  =  ^'        ^^Hia)  ^  =  ^'  ^^Hcb) 

C=D,       (2)}^"^  A+C=B  +  D,     (4)1^"^ 

and  let  (4)  be  obtained  by  adding  the  corresponding  mem- 
bers of  (1)  and  (2) ;  we  are  to  prove  that  system  (b)  is  equiv- 
alent to  system  (a). 

Any  solution  of  (a)  renders  (1)  and  (2)  identities.  But, 
if  (1)  and  (2)  are  identities;  by  §  32,  (3)  and  (4)  are  iden- 
tities ;  hence  any  solution  of  system  (a)  is  a  solution  of 
system  (b). 

Conversely,  any  solution  of  (6)  renders  (3)  and  (4)  iden- 
tities. But,  if  (3)  and  (4)  are  identities ;  by  §  32,  (1)  and  (2) 
are  identities ;  hence  any  solution  of  system  (b)  is  a  solution 
of  (a). 

Hence  (a)  and  (6)  are  equivalent  systems. 

In  like  manner  the  theorem  could  be  proved  if  the  sys- 
tems (a)  and  (b)  contained  three  or  more  equations. 


SYSTEMS   OF  LINEAR  EQUATIONS 


209 


The  foregoing  examples  illustrate  the  following  rule  for 
elimination  by  addition  or  subtraction : 

Reduce  the  equations  to  the  form  ax  -\-by  =  c. 

Find  the  L.  C.  M.  of  the  coefficients  of  the  unknown  to  he 
eliminated.  Multiply  both  members  of  each  equation  by  the 
quotient  of  this  L.  C.  M.  divided  by  the  coefficients  of  that  un- 
known in  the  equation. 

Add  or  subtract  the  corresponding  members  of  the  equations 
thus  denved,  according  as  the  coefficients  of  the  unknown  to  be 
eliminated  are  opposite  or  like  in  quality. 


Exercise  81. 


Solve  each  of  the  following  systems  by  the 
addition  or  subtraction : 


1.  3  a; +  4  2/ =  10, 
4:X  +  y  =  9. 

2.  x-\-2y  =  lS, 
3x-{-y  =  li. 

3.  2x  —  y  =  9j 
Sx-7y  =  19. 

4.  4  a; +  7  2/ =  29, 
x-{-Sy  =  ll. 

6.   2x  +  y  =  10,     1 
7  a;  +  8  2/  =  53.  J 

6.  5  a; +  6  2/ =  17, 
6  a;  4-  5  y  =  16 

7.  8x-2/  =  34, 1 
x  +  Sy  =  5S.} 


;;1 


8.   15  a; +7  2^  =  29, 
9  aj  +  15  2/  =  39, 


:1 


9.   14:X-Sy  = 
6  a;  +  17  2/  = 

10.  28  a: -23  2/ 
63  a;  -  25  2^ 

11.  S5x-{-17y 
56x-lSy 

12.  5  a;  — 7  2/  = 
7x  +  5y  = 

13.  15  a; +  77  2/ 
55x-S3y 

14.  5x  =  7y  — 
21x-9y  = 

15.  6y  —  5x  = 
12x-9y  = 

16.  21  a; -50  2/ 
2Sx-27y 


method  of 

=  39,1 
=  36.1 

=  33,   1 
=  101.  J 

=  86,1 
=  17.1 


u] 


0, 

74 

=  92, 

f 

21, 

=  7{ 

18,1 
=  0.1 

=  60,   I 
=  199.  J 


=  92,1 
=  22.1 


..) 


210 


ELEMENTS   OF  ALGEBBA 


205.  The  following  example  illustrates  a  special  form 
of  the  method  of  elimination  by  addition,  which  is  called 
elimination  by  undetermined  multipliers. 


(3) 


(4) 


5/3. 


Ex.    Solve  the  system      3  x  —  5  </  =  2, 

Multiplying  (1)  by  an  arbitrary  multiplier  /c,  we  obtain 

3  kx  -  5  ky  =  2k. 

Adding  (3)  and  (2),  we  obtain 

(3  k  +  [>')x  -  {5  k -\-  2)y  =  2k-\-  10. 

Putting  the  coefficient  of  x  in  (4)  equal  to  0,  we  obtain  k  =  - 

Substituting  —  5/3  for  k  in  (4),  we  obtain  y  =  2.  (5) 

Putting  the  coefficient  of  y  in  (4)  equal  to  0,  we  obtain  A:  =  —  2/5. 

Substituting  —  2/5  for  k  in  (4),  we  obtain  x  =  4.  (6) 

The  two  equations,  y  ^2  and  x  =  4,  which  result  from  the  two 
different  values  of  k  in  (4)  form  a  system  which  by  (i)  and  (iii)  is 
equivalent  to  (a). 

No  one  method  of  elimination  is  preferable  for  all  cases. 
The  learner  should  aim  to  select  that  method  which  is  best 
suited  to  the  system  to  be  solved. 


Exercise  82. 


Solve  each  of  the  following  systems  by  that  method  which 
is  best  suited  to  it : 


1.  57  a; +  25^  =  3772, 
25  0^4-57  2/ =  1148. 

2.  93a;H-15i/  =  123, 
15x4- 93  2/ =  201. 

3.  Wx-hl9y  =  lS,] 


5. 


+  52/  =  -4, 


5 


SYSTEMS   OF  LINEAR  EQUATIONS 


211 


6.    ''-±l-^iy  =  2, 

y  +  11      x  +  l^ 
11  2 


1. 


2x  —  5y     x-^7  _  ^ 


8. 


3 

a:-2 


4 


2a;-5     ll-2y 


=  0.J 


9.    ^-3^  =  0, 


2x 


7      13 -y 
3  16 


0. 


10.   ^-f  =  4, 


11.  2a;-f2/=:0, 
i2/-3a;  =  8.J 

12.  \x-\-\y  =  n, 


13.  3. r  — 7  2^  =  0, 
f  a;  +  |2/=7. 

14.  lx-\y  =  0, 
3 


x  +  ^y  =  ll,] 


15.  aa; -h  fty  =  (a  +  6)2 
ax  —  by=  a^  —  Ir. 

16.  «x  +  &y=a24-6V 
hx-{-ay  =  2db. 

17.  aa;  +  6y  =  a--62, 
6a; 


+  6y  =  a2-62  , 
+  a?/  =  a2_62  J 


18.  x  +  y  =  a-^h, 
ax  —  by  =  b^  —  a'. 

19.  b-x-a'y  =  0, 
bx 


4-a?/=:aH-6.    J 


20.  x  —  y  =  a~b, 
ax-by  =  2a^-2b-.  J 

21.  ax-by  =  a'  +  b^,) 
x+  y  =  2a.  j 

22.  6a;  —  a?/ =  ft'',  I 
ax  —  by  =  al  J 

23.  aa;H-6?/  =  l,  | 
6a;  +  ay  =  l.  J 


24.    (a  +  b)x-(a-b)y  =  3ab,] 
(a  +  6)2/ -  (a  -  6)a;  =  ab.     J 


a^a;  +  6^y  =  c*.  J 


26.    ^  + 


a      6      a6' 

a?  _    y^   1 
a'     6'     a'6'" 


212 


ELEMENTS   OF  ALGEBRA 


27. 

Sx_^2y 

a        b 
9x_6y_^ 

a        b 

28. 

qx  —  rb  =p{a  —  y),  " 

^+,.=,(1+1). 

29. 

-^-1  =1. 

m'      m 

30. 

a     b 

3a     66     3  J 

31. 

a     0 

a'     V       J 

32. 

b     a      b 

206.  Two  conditions  are  said  to  be  consistent  or  inconsistent 
according  as  they  can  or  cannot  be  satisfied  at  the  same 
time. 

Equations  which  express  consistent  conditions  and  there- 
fore have  one  or  more  solutions  in  common  are  called  con- 
sistent equations.  Thus  the  equations  in  any  of  the  above 
systems  are  consistent  equations. 

Equations  which  express  inconsistent  conditions  and  there- 
fore have  no  solution  in  common  are  called  inconsistent 
equations. 


E.g..,  the  equations 


3  5c  +  3y  =  15, 


(1) 
(2) 


express  inconsistent  conditions  and  have  no  solution  in  common.    For 
if  a;  -I-  2/  is  4,  3(x  +  ?/)  is  12  and  cannot  therefore  be  15. 


207,  Each  of  the  foregoing  systems  of  linear  equations 
illustrates  the  following  theorem : 

If  the  two  equations  of  a  system  in  two  unknowns  are  linear, 
independent,  and  consistent,  the  system  has  one,  and  only  one, 
solution. 


SYSTEMS  OF  LINEAR  EQUATIONS  213 

Proof.  By  the  principles  of  equivalent  equations  and  (i) 
in  §  201,  any  system  of  two  linear  equations  can  be  reduced 
to  an  equivalent  system  of  the  form 


)(«) 


ax-\-hy  =  c,  (1) 

a'x  +  h'y  =  c'.  (2) 

Multiply  (1)  by  h',  ab'x  +  bb'y  =  b'c.  (3) 

Multiply  (2)  by  b,  a'bx  +  bb'y  =  be',  (4) 

Subtract  (4)  from  (3),  (ab'  -  a'b)  x  =  b'c  -  be'.  (5)  ^ 

From  (1),  ax-\-by  =  c.  (1)  J 

By  (i)  and  (iii),  system  (6)  is  equivalent  to  system  (a). 

When  ab'  —  a'b  =  0  and  b'c  —  be'  =  0,  (5)  is  an  identity, 
and  system  (b)  or  (a)  has  all  the  solutions  of  equation  (1) ; 
hence  equations  (1)  and  (2)  are  equivalent. 

When  ab'  —  a'b  =  0  and  b'c  —  be'  ^  0,  no  value  of  x  will 
satisfy  (5) ;  hence  system  (6)  or  (a)  has  no  solution,  and 
equations  (1)  and  (2)  are  inconsistent. 

Hence  (1)  and  (2)  are  not  independent  and  consistent 
unless 

ab'  -  a'b  ^  0. 

When  ah'  —  a'b  ^  0,  x  has  one,  and  only  one,  value  in 
(5),  and  this  value  of  x  will  give  one,  and  only  one,  value 
for  y  in  (1) ;  hence  system  (6)  or  (a)  has  one,  and  only  one, 
solution. 

When  ab'  —  a'b  ^  0,  from  (5)  we  obtain 

X  =  (b'c  -  be')  /  (ab'  -  a'b).  (6) 

Similarly,        y  =  (ae'  —  a'e)  /  (ab'  —  a'b).  (7) 

208.   Systems  of  three  linear  equations. 

Ex.  1.   Solve  6x  +  2y-5z  =  lS,  (1)  1 

3a;  +  3y-20=13,  (2)  i  (a) 

1x  +  5y-Sz  =  26.  (3)  I 


(ft) 


(0 


214  ELEMENTS   OF  ALGEBRA 

To  eliminate  y,  we  can  proceed  as  follows : 

Multiply  (1)  by  3,       \^x  +  Qy  -\^z  =  39. 

Multiply  (2)  by  2,         Qx-^Qy  -    4:Z  =  2Q. 

Subtract,  12  x  -  11  2;  =  13.  (4) 

Multiply  (1)  by  5,     30  a:  +  10  «/  -  25  ^  =  65. 

Multiply  (3)  by  2,     Ux-{-10y-    6z  =  52. 

Subtract,  16  x  -  19  0  =  13.  (5) 

Solving  system  (6),  i.e.,  (4)  and  (5),  we  obtain 

z=l,  (6)  I 

x  =  2.  (7)  J 

From  (6),  (7),  and  (1),  y  =  3.  (8) 

The  systems  (6)  and  (c)  are  equivalent. 

But  (b)  with  (1)  forms  a  system  equivalent  to  (a);  hence  (c)  with 
(1),  or  (c)  with  (8),  forms  a  system  equivalent  to  (a). 
Hence  the  solution  of  system  (a)  is  2,  3,  1. 

Ex.  2.    Solve  3  X  +  2  y  +  4  2:  =  19,  (1)1 

2x+ 5^4-30  =  21,  (2)  K«) 

Sx-y-\-z  =  i.  (3)  J 

From  (3),  y  =  Sx  +  z-i.  (4) 

Substituting  in  (1)  and  (2)  the  value  of  y  in  (4),  we  obtain 
3  X  +  2(3  X  +  0  -  4)  +  4  2:  =  19, 
and  2x4-5(3x  +  ;3 -4)+32  =  21; 

or,  9  X  -i-  6  2;  =  27,  | 

and  17x  + 8^  =  41.  J 

Solving  system  (6),  we  obtain 

(5)1 

X  =  1.  (6)  J  '  ' 

From  (4),  (5),  (6),  y  =  2.  (7) 

By  (ii),  system  (6)  with  (4)  forms  a  system  equivalent  to  (a); 
hence  (c)  with  (4),  or  (c)  with  (7),  forms  a  system  equivalent  to  (a). 
Hence  the  solution  of  (a)  is  1,  2,  3. 


SYSTEMS   OF  LINEAR  EQUATIONS 


215 


The  foregoing  examples  illustrate  the  following  method 
of  solving  a  system  of  three  linear  equations : 

From  any  two  of  the  three  equations  derive  an  eq^iation, 
eliminating  an  unknown;  next  from  the  third  equation  and 
one  of  the  other  two  derive  a  second  equation  eliminating  the 
same  unknown. 

Solve  for  these  two  unknowns  the  two  equations  thus  derived, 
and  substitute  the  values  of  these  two  unknowns  in  the  simplest 
equation  which  contains  the  third  unknown. 

209.  From  a  system  of  four  linear  equations  we  can  elimi- 
nate one  of  the  four  unknowns,  and  thiis  obtain  a  new  system 
with  three  unknowns.  Solving  this  new  system,  we  can 
substitute  the  values  thus  obtained  in  the  simplest  equation 
which  contains  the  fourth  unknown. 


Solve 


Exercise  83. 


1.  .T-f  .3?/ -1-42  =  14,] 
or +  2?/ -1-2  =  7, 
2x-\-y-\-2z=2. 

2.  a; -1-22/ +  22  =  11, 
2x  +  y-\-z  =  l, 
3a;  +  42/-h2  =  14. 

3.  ^x-2y-\-z  =  2,] 
2aj-j-32/-2  =  5, 
.T  +  2/  -h  2  =  6. 

4.  x-\-y^z  =  l, 
2a;-|-3  2/  +  2  =  4, 
4  a;  H-  9  2/  +  2  =  16. 


5.  5x  +  3y-\-7z  =  2y 
2a; -42/ +  92  =  7, 
3x  +  22/  +  62  =  3. 

6.  x-{-2y-3z  =  6, 
2.T  +  42/-72  =  9, 
3a;  —  ^  —  52  =  8. 

7.  x-2y  +  3z  =  2, 
2a;  — 3  2/ +  2  =  1, 
3x-y-\-2z  =  9. 


8.    3x-\-2y-z  =  20, 
2x-\-3y-\-6z  =  70 
a;  —  2/  +  62  =  41.     J 


216 


ELEMENTS  OF  ALGEBRA 


9.  2a; +  3?/ +  42  =  20, 
3a; +  42/ +  5^  =  26, 
Sx-{-5y-{-6z  =  31. 

10.  Sx-4:y  =  6z-16, 
4:X-y  =  z-\-5, 

a;  =  32/ +  2(2-1).    J 

11.  ax-\-by  =  l, 
by  -\-  cz  =  1, 
cz  +  ax  =  1. 

12.  cy  -\-hz  —  he, 
az  -\-  ex  =  ca, 
hx  -\-  ay  =  ah. 

13.  a;-|  =  6, 


y-- 

^      7 


=  8.    !- 


=  10. 


14.    ^(a;+2!-5)=2/-2, 
i(a;+;2_5)=2a;-ll, 
2a;-ll  =  9-(a;+22). . 


15. 


16. 


17. 


18. 


19. 


20. 


a; +  20  =  12/ +  10, 
a;  +  20  =  2  2  +  5, 
22  +  5  =  110 -(2/ +  2). 
ax  +  &2/  =  Ij  1 

62/  +  C2  =  1,   I 

C2  +  aa;  =  1.  J 
cy  -i-bz  =  be, 
az  -^  ex  =  ca, 
bx  -\-  ey  =  ab. 
X  —  ay  -\-  a-z  =  a^, 
X  —by  -\-  bh  =  b^, 
X  —  cy  +  c^2  =  c^.  J 

•'c  +  2/  +  ^  "~  "^  =  H> 

a;  +  2/  -  2  +  w  =  17, 

a;  —  2/  +  2  +  w=    9, 

■  X  -\-  y  -\-  z  -\-  u  =  12. 

x-\-y  -{-z  =  % 
x-\-y-{-u  =  l, 
a;  +  2  + 1<  =  8, 
2/  +  2  +  ?^  =  9.  J 


SYSTEMS  OF  FRACTIONAL  EQUATIONS. 

210.  In  clearing  of  fractions  the  equations  of  a  system, 
no  solution  will  be  lost,  but  new  solutions  may  be  introduced 
even  when  we  clear  of  fractions  in  the  simplest  manner. 

Ex.  1.    Solve  the  system  4  a;  —  2  ?/  =2, 


5x  +  l 


3y-l      8 
Clearing  (2)  of  fractions  and  transposing,  we  have 
40  a: -33?/ =-19. 


0) 

(2) 
(3) 


(a) 


SYSTEMS  OF  LINEAR  EQUATIONS  217 

The  solution  of  system,  (1)  and  (3),  is  2,  3. 

In  clearing  (2)  of  fractions  we  multiplied  by  the  unknown  factor 
3  y  —  1 ;  hence  any  solution  which  was  introduced  will  be  a  solution 
of  the  equation  3y  —  1=0,  or3y  +  0x  —  1  =  0. 

Since  2,  3  is  not  a  solution  of  this  equation,  it  was  not  introduced 
in  clearing  (2)  of  fractions. 

Hence  2,  3  is  the  one  and  only  solution  of  system  (a). 


Ex.  2.   Solve  the  system      5 «  —  y  =  2,  (1) 

^      4-^^  =  0.  (2) 


(a) 


x-1      y-S 
Clear  (2)  of  fractions,  z-\-y  =  4.  (3) 

The  solution  of  system,  (1)  and  (3),  is  1,  3. 

To  clear  (2)  of  fractions  we  multiplied  by  the  unknown  factor 
(a;— l)(y— 3),  and  1,  3  is  a  solution  of  the  equation  (x— l)(y— 3)  =  0. 

Hence  the  solution  1,  3  may  have  been  introduced  by  clearing  (2) 
of  fractions. 

By  trial  we  find  that  1,3  is  not  a  solution  of  (2)  ;  hence  the  solu- 
tion 1,  3  was  introduced,  and  system  (a)  has  no  solution  ;  that  is,  its 
equations  are  inconsistent. 

211.  A  system  of  fractional  equations  which  are  linear  in 
the  reciprocals  of  their  unknowns  is  readily  solved  without 
clearing  of  fractions,  by  treating  these  reciprocals  as  the 
unknowns. 


(«) 


Ex.  1.   Solve  the  system             a/x  +  c/y  = 

:m, 

(1) 

b/x  +  d/y  = 

:  n. 

(2) 

Multiply  (1)  by  6,          ab(l/x)  +  cbil/y)  = 

:  bm. 

(3) 

Multiply  (2)  by  a,           ab(\/x)  +  ad(\/y)  = 

--  an. 

(4) 

Subtract  (4)  from  (3) ,         (6c  -  ad)  (1  /y)  = 

:  6m- 

-  an. 

.'.  y  = 

6c- 

bm  - 

-ad 
-an 

(5) 

Multiply  (1)  by  d,          ad(\/x)  +  cd(\/y)  = 

-  dm. 

(6) 

Multiply  (2)  by  c,            6c(l/x)  +  cd(l/?/)  = 

=  C7l. 

(7) 

Subtract  (7)  from  (6),         {ad  -  6c)(l/x)  = 

-  dm  - 

-  en. 

.'.  x  = 

ad- 

-be 

(S) 

dm 


218 


ELEMENTS   OF  ALGEBRA 


Multiplying  (1)  and  (2)  by  xy  to  clear  them  of  fractions  would  give 
us  a  system  of  quadratic  equations  and  introduce  the  new  solution  0,  0. 


Ex.  2.    Solve  the  system 


36, 


1  +  1  +  1 
X     tj     z 


1  +  5-1  =  28, 

X     y     z 


^y     2z 


20. 


Subtract  (1)  from  (2),  2  (\/y)-2  (l/z)  =  -S, 
Subtract  (3)  from  (1),  ^  (l/y)+ l(l/z')=  W. 
Solving  system  (6),  y  =  1/12, 

0  =  1/16. 
From  (1),  (6),  and  (7),  x  =  1/8. 


(1) 

(^) 
(3) 


4)  J 

5)  J 


(4) 

( 

(6) 

(7) 

(8) 


(a) 


(&) 


Solve  the  system 

Exercise  84. 

1.   «-»  =  !, 

X     y 

5. 

^  +  1«  =  79, 

X      y 

15+6^7. 

X      y 

15-1  =  44. 

a;      ?/ 

2.  5_4=2, 

X     y 

6. 

6     7      „ 

18  +  18  =  10. 
X        y 

2  +  11  =  3. 

^     y 

3..   5_5  =  9j 

a;     y 

7. 

^  +  2  -7 1 

1-2  =  5. 
X     y 

7         1    -3. 
6*     10?^         J 

4.   5  +  ^  =  3, 

a;     y 

8. 

14-2-3     1 
2X  +  32,-'' 

5-^  =  1 

4a;     oy 

9. 

SYSTEMS   OF  LINEAR   EQUATIONS 


219 


9.   -  +-  =  a, 
X      y 

X     y 


10. 


m     n 


mr 


11. 


X  y  n 
7i_m_  nr 
X      y      m 


mx     ny 
mx     ny 


12.  -  +  f  =  2, 
ax     by 

—  --  =  7. 
ax     by 

13.  A_2  ^4, 

ax     by 

a;     2^ 


14. 1 =  m  +  71, 

?ia;     ?n.y 

X      y 


1       2 
15.    i_-_j_4  =  0, 
a;     ^ 

1-1  +  1=0, 

2/     2 


5  +  3 

2!      a; 


14. 


16.    1  +  1  +  1 

X      y      z 


36, 


1  +  ^-1  =  28, 
X     y     z 

a;     3y     2z 


17.    -^  +  -^ 


J_      J J^^l 

2x~^4.y     3z     4' 

1^  1^ 
X     Sy 

i-^  +  i  =  2A. 
X     by     z 


18.    3a; +  4^  =  11, 

3y  +  l       7* 


19. 


2x-\-A 

=  1, 
=  2. 

3y-l 
lx-2 

.V  +  3 

20. 


x±2y±J^^ 
Sx-\-y-l       ' 

3x+x±l  =  i 
4a;-2/-2 


21. 


5a;  +  2y     9a;  +  4y 

2a;     ^2 
7a;  +  2/      5* 
22.    5x  —  Sy  =  S, 

1     +^-  =  0. 


a;  —  3      y  —  4: 


CHAPTER  XV 
PROBLEMS   SOLVED  BY  SYSTEMS 

212.  A  determinate  jrroblem  is  one  which  has  a  finite  num- 
ber of  solutions.  Every  determinate  problem  must  contain 
as  many  independent  consistent  conditions,  expressed  or 
implied,  as  unknown  numbers.  If  in  any  such  problem  we 
denote  each  unknown  by  a  letter,  and  express  each  condition 
by  an  equation,  we  shall  obtain  as  many  independent  con- 
sistent equations  as  there  are  unknowns. 

The  solutions  of  the  system  of  equations  thus  obtained 
will  give  the  solutions  of  the  problem. 

Prob.  1.   Find  two  numbers  such  that  twice  the  greater  exceeds  three 
times  the  less  by  6,  and  that  twice  the  less  exceeds  the  greater  by  2. 
Let  X  =  the  greater  number,  and  y  =  the  less. 
Then,  by  the  first  condition,  we  have 

2x-Sy  =  Q,  (1). 

and  by  the  second  condition  we  have  \  (a) 

2y-x  =  2.  (2)  J 

From  system  (a),  x  =  18,  the  greater  number ; 
and  y  =  10,  the  less  number* 

Prob.  2.  A  number  expressed  by  two  digits  is  equal  to  six  times  the 
sum  of  its  digits,  and  the  digit  in  the  tens'  place  is  greater  by  one  than 
the  digit  in  the  units'  place.     Find  the  number. 

Let  X  =  the  digit  in  tens'  place, 

and  y  =  the  digit  in  units'  place. 

220 


PROBLEMS  221 

Then,  from  the  first  condition,  we  have 

10x  +  i/  =  C(x  +  ?/),  (1)- 

and  from  the  second  condition  we  have  ■  (a) 

x-y  =  l.  (2)  . 

From  system  (a),  a;  =  5,  the  digit  in  tens'  place ; 
and  2/  =  4,  the  digit  in  units'  place. 

That  is,  the  required  number  is  54. 

Prob.  3.  If  the  numerator  of  a  fraction  is  increased  by  2  and  the 
denominator  by  1,  it  becomes  equal  to  5/8,  and  if  the  numerator  and 
denominator  are  each  diminished  by  1,  it  becomes  equal  to  1/2.  Find 
the  fraction. 

Let  X  =  the  numerator,  and  y  =  the  denominator  ;  then, 

from  the  first  condition,  ^-±-?  =  -,  (1)  ] 

and  from  the  second,  - — -  =  -•  (2)  J 

The  solution  of  sy.stem  (a)  is  8,  15  ;  hence  the  fraction  is  8/15. 

Prob.  4.  A  man  and  a  boy  can  do  in  15  days  a  piece  of  work  which 
would  be  done  in  2  days  by  7  men  and  9  boys.  How  long  would  it 
take  one  boy  or  one  man  to  do  it. 

Let  X  =  the  number  of  days  it  would  take  one  man  to  do  the  whole 
work,  and  y  =  the  number  of  days  it  would  take  one  boy. 

Let  the  whole  work  be  represented  by  1. 

Then  in  one  day  a  man  would  do  1/x  of  the  work,  and  a  boy  l/y 
of  it. 

Hence,  by  the  first  condition,  we  have 

15/x  +  Vo/y  =  1,  (1)  ^ 

and  by  the  second  condition  we  have  [  (a) 

14/x  +  18/?/  =  1.  (2)  J 

The  solution  of  system  (a)  is  20,  60. 

Hence  one  man  would  do  the  work  in  20  days,  and  one  boy  in  60 
days. 


222  ELEMENTS   OF  ALGEBRA 

Exercise  85. 

1.  Six  horses  and  7  cows  can  be  bought  for  $1250,  and 
13  cows  and  11  horses  can  be  bought  for  $2305.  Find  the 
value  of  each  animal. 

2.  Four  times  B's  age  exceeds  A's  age  by  20  years,  and 
\  of  A's  age  is  less  than  B's  age  by  2  years.  Find  their 
ages. 

3.  Find  a  fraction  such  that  if  1  be  added  to  its  denomi- 
nator it  reduces  to  |-,  and  if  2  be  added  to  its  numerator  it 
reduces  to  f . 

4.  A  man  being  asked  his  age,  replied :  "  If  you  take 
2  years  from  my  present  age  the  result  will  be  double  my 
wife's  age,  and  3  years  ago  her  age  was  \  of  what  mine  will 
be  in  12  years."     Find  their  ages. 

5.  One-eleventh  of  A's  age  is  greater  by  2  years  than  \ 
of  B's,  and  twice  B's  age  is  equal  to  what  A's  age  was  13 
years  ago.     Find  their  ages. 

6.  In  8  hours  A  walks  12  miles  more  than  B  does  in 
7  hours ;  and  in  13  hours  B  walks  7  miles  more  than  A  does 
in  9  hours.     How  many  miles  does  each  walk  per  hour  ? 

7.  At  an  election  the  majority  was  1G2,  which  was  ^\  of 
the  whole  number  of  voters.  What  was  the  number  of  the 
votes  on  each  side  ? 

8.  A  and  B  have  $  250  between  them  ;  but  if  A  were  to 
lose  half  his  money^  and  B  |  of  his,  they  would  then  have 
only  $  100.     How  much  has  each  ? 

9.  A  man  bought  8  cows  and  50  sheep  for  $  1125.  He 
sold  the  cows  at  a  profit  of  20%,  and  the  sheep  at  a  profit 
of  10%,  and  received  in  all  $1287.50.  What  was  the  cost 
of  each  cow  and  of  each  sheep  ? 

10.  Twenty-eight  tons  of  goods  are  to  be  carried  in  carts 
and  wagons,  and  it  is  found  that  this  will  require  15  carts 
and  12  wagons,  or  else  24  carts  and  8  wagons.  How  much 
can  each  cart  and  each  wagon  carry  ? 


PROBLEMS  223 

11.  A  and  B  can  perform  a  certain  task  in  30  days,  work- 
ing together.  After  12  days,  however,  B  was  called  off,  and 
A  finished  it  by  himself  24  days  after.  How  long  would 
each  take  to  do  the  work  alone  ? 

12.  Find  the  fraction  such  that  if  you  quadruple  the 
numerator  and  add  3  to  the  denominator  the  fraction  will 
be  doubled,  but  if  you  add  2  to  the  numerator  and  quadruple 
the  denominator,  the  fraction  will  be  halved. 

13.  The  first  edition  of  a  book  had  600  pages,  and  was 
divided  into  two  parts.  In  the  second  edition  J  of  the 
second  part  was  omitted  and  30  pages  were  added  to  the 
first  part.  The  change  made  the  two  parts  of  the  same 
length.  How  many  pages  were  in  each  part  in  the  first 
edition  ? 

14.  A  marketman  bought  eggs,  some  at  3  for  5  cents,  and 
some  at  4  for  5  cents,  and  paid  for  all  $  5.60;  he  afterwards 
sold  them  at  24  cents  a  dozen,  clearing  $  1.80.  How  many 
eggs  did  he  buy  at  each  price  ? 

15.  In  a  bag  containing  black  and  white  balls,  half  the 
number  of  white  is  equal  to  a  third  of  the  number  of  black ; 
and  twice  the  whole  number  of  balls  exceeds  3  times  the 
number  of  black  balls  by  4.  How  many  balls  does  the  bag 
contain  ? 

16.  A  crew  that  can  row  10  miles  an  hour  down  a  river, 
finds  that  it  takes  twice  as  long  to  row  up  the  river  as  to 
row  down.     Find  the  rate  of  the  current. 

17.  A  certain  number  between  10  and  100  is  8  times  the 
sum  of  its  digits,  and  if  45  be  subtracted  from  it  the  digits 
will  be  reversed.     Find  the  number. 

18.  If  A  were  to  receive  $  50  from  B,  he  would  then  have 
twice  as  much  as  B  would  have  left;  but  if  B  were  to  receive 
$50  from  A,  B  woukl  have  3  times  as  much  as  A  would 
have  left.     How -much  has  each? 


224  ELEMENTS  OF  ALGEBRA 

19.  A  farmer  sold  30  bushels  of  wheat  and  50  bushels  of 
barley  for  $93.75.  He  also  sold  at  the  same  prices  50 
bushels  of  wheat  and  30  bushels  of  barley  for  $  96.25. 
What  was  the  price  of  the  wheat  per  bushel  ?    , 

20.  One  rectangle  is  of  the  same  area  as  another  which 
is  6  yards  longer  and  4  yards  narrower;  it  is  also  of  the 
same  area  as  a  third,  which  is  8  yards  longer  and  5  yards 
narrower.     What  is  the  area  of  each  ? 

21.  A  boy  rows  8  miles  with  the  current  in  1  hour  4  min- 
utes, and  returns  against  the  current  in  2|-  hours.  At  what 
rate  would  he  row  in  still  water  ?  What  is  the  rate  of  the 
current  ? 

22.  A,  B,  C,  D  have  $1450  among  them;  A  has  twice 
as  much  as  C,  and  B  has  3  times  as  much  as  D ;  also  C  and 
D  together  have  $  250  less  than  A.  Find  how  much  each 
has. 

23.  A,  B,  C,  D  have  $1350  among  them ;  A  has  3  times 
as  much  as  C,  and  B  5  times  as  much  as  D ;  also  A  and  B 
together  have  $  250  less  than  8  times  what  0  has.  Find 
how  much  each  has. 

24.  A  number  consists  of  2  digits  followed  by  zero.  If 
the  digits  be  interchanged,  the  number  will  be  diminished 
by  180 ;  if  the  left-hand  digit  be  halved,  and  the  other  digit 
be  interchanged  with  zero,  the  number  will  be  diminished 
by  454.     Find  the  number. 

25.  A  train  travelled  a  certain  distance  at  a  uniform  rate ; 
had  the  speed  been  6  miles  an  hour  more,  the  journey  would 
have  occupied  4  hours  less ;  and  had  the  speed  been  6  miles 
an  hour  less,  the  journey  would  have  occupied  6  hours  more. 
Find  the  distance. 

Let  X  =  the  number  of  miles  the  train  runs  per  hour, 

and  y  =  the  number  of  hours  the  journey  takes. 

Then  xi/ =(x  +  6)(?/ -  4),  ] 

and  xy=ix-Q)(y  +  6).\ 


PROBLEMS  225 

26.  A  traveller  walks  a  certain  distance ;  had  he  gone  ^ 
mile  an  hour  faster,  he  would  have  walked  it  in  ^  of  the 
time ;  had  he  gone  ^  mile  an  hour  slower,  he  would  have 
been  2i  hours  longer  on  the  road.     Find  the  distance. 

27.  A  man  walks  35  miles,  partly  at  the  rate  of  4  miles 
an  hour,  and  partly  at  5 ;  if  he  had  walked  at  5  miles  an 
hour  when  he  walked  at  4,  and  vice  versa,  he  would  have 
covered  2  miles  more  in  the  same  time.  Find  the  time  he 
was  walking. 

28.  A  fishing-rod  consists  of  two  parts ;  the  length  of  the 
upper  part  is  f-  that  of  the  lower  part ;  and  9  times  the  upper 
part  together  with  13  times  the  lower  part  exceeds  11  times 
the  whole  rod  by  36  inches.    Find  the  lengths  of  the  two  parts. 

29.  A  man  put  $12,000  at  interest  in  three  sums,  the 
first  at  5  per  cent,  the  second  at  4  per  cent,  and  the  third 
at  3  per  cent,  receiving  for  the  whole  $  490  a  year.  The 
sum  at  5  per  cent  is  half  as  much  as  the  other  two  sums. 
Find  each  of  the  three  sums. 

30.  A,  B,  and  C  can  together  do  a  piece  of  work  in  30 
days ;  A  and  B  can  together  do  it  in  32  days ;  B  and  C  can 
together  do  it  in  120  days.  Find  the  time  in  which  each 
alone  could  do  the  work. 

31.  A  certain  company  in  a  hotel  found,  when  they 
came  to  pay  their  bills,  that  if  there  had  been  3  more  per- 
sons to  pay  the  same  bill,  they  would  have  paid  f  1  each 
less  than  they  did;  and  if  there  had  been  2  fewer  persons, 
they  would  have  paid  $  1  each  more  than  they  did.  Find 
the  number  of  persons,  and  the  number  of  dollars  each  paid. 

32.  A  railway  train,  after  travelling  1  hour,  is  detained 
30  minutes,  after  which  it  proceeds  at  f  of  its  former  rate, 
and  arrives  20  minutes  late.  If  the  detention  had  occurred 
10  miles  farther  on,  the  train  would  have  arrived  5  minutes 
later  than  it  did.  Find  the  first  rate  of  the  train,  and  the 
distance  travelled. 


226  ELEMENTS   OF  ALGEBRA 

Let  X  =  the  number  of  miles  the  train  at  first  ran  per  hour  ; 

and  y  =  the  number  of  miles  in  the  whole  distance  travelled. 

Then     y  —  x  =  the  number  of  miles  to  be  travelled  after  the  de- 
tention, 

=  the  number  of  hours  required  to  travel  y  —  x  miles 
at  the  rate  before  the  detention, 

and       ^^^ — -^^  =  the  number  of  hours  required  to  travel  y  —  x  miles 
at  the  rate  after  the  detention. 


y 

—  X 

X 

4(2/- 

-X) 

-X     4(y  -x)  , 
c              5x 

_io 

go' 

Hy-x- 

10) 

5 

5a; 

60 

10_ 

X 

40. 
5x 

_  5 
60" 

.-.  x  =  24 

t,  y- 

=  44. 

Hence  ^Lr^  _  li^^^_2  =  i!£.  (i) 

X  5x  CO  ■  ^  ^ 

Similarly,    y  -  x  -  10  _  4(y -^x  -  10)  ^  |_^  ^2^ 
Subtract  (2)  from  (1), 


Hence  the  first  rate  was  24  miles  an  hour,  and  the  distance  travelled 
was  44  miles. 

33.  A  railway  train,  after  travelling  1  hour,  meets  with 
an  accident  which  delays  it  1  hour,  after  which  it  proceeds 
at  I  of  its  former  rate,  and  arrives  at  the  terminus  3  hours 
behind  time  ;  had  the  accident  occurred  50  miles  farther  on, 
the  train  would  have  arrived  1  hour  20  minutes  sooner. 
Find  the  length  of  the  line,  and  the  original  rate  of  the 
train.  Ans.  100  miles,  25  miles  per  hour. 

34.  A  jockey  has  2  horses  and  2  saddles.  The  saddles 
are  worth  $  15  and  $  10  respectively.  The  value  of  the 
better  horse  and  better  saddle  is  |  that  of  the  other  horse 
and  saddle ;  and  the  value  of  the  better  saddle  and  poorer 
horse  is  |f  that  of  the  other  horse  and  saddle.  Find  the 
worth  of  each  horse. 

35.  Five  thousand  dollars  is  divided  among  A,  B,  C, 
and  D.  B  gets  half  as  much  as  A ;  the  excess  of  C's  share 
over  D's  share  is  equal  to  ^  of  A's  share,  and  if  B's  share 


PROBLEMS  227 

■were  increased  by  $  500  he  would  have  as  much  as  C  and  D 
have  between  them.     Find  how  much  each  gets. 

36.  A  i^arty  was  composed  of  a  certain  number  of  men 
and  women,  and,  when  4  of  the  women  were  gone,  it  was 
observed  that  there  were  left  just  half  as  many  men  again 
as  women ;  they  came  back,  however,  with  their  husbands, 
and  now  there  were  only  a  third  as  many  men  again  as 
women.     What  was  the  original  number  of  each  ? 

37.  Two  vessels  contain  mixtures  of  wine  and  water;  in 
one  there  is  3  times  as  much  wine  as  water,  in  the  other 
5  times  as  much  water  as  wine.  Find  how  much  must  be 
drawn  off  from  each  to  fill  a  third  vessel  which  holds  7  gal- 
lons, in  order  that  its  contents  may  be  half  wine  and  half 
water. 

38.  There  is  a  number  of  3  digits,  the  last  of  which  is 
double  the  first ;  when  the  number  is  divided  by  the  sum  of 
the  digits,  the  quotient  is  22 ;  and  when  by  the  product  of 
the  last  two,  11.     Find  the  number. 

39.  Some  smugglers  found  a  cave  which  would  exactly 
hold  the  cargo  of  their  boat ;  viz.  13  bales  of  silk  and  33 
casks  of  rum.  While  unloading,  a  revenue  cutter  came 
in  sight,  and  they  were  obliged  to  sail  away,  having  landed 
only  9  casks  and  5  bales,  and  filled  J  of  the  cave.  How 
many  bales  separately,  or  how  many  casks,  would  it  contain  ? 

40.  There  are  2  alloys  of  silver  and  copper,  of  which  one 
contains  twice  as  much  copper  as  silver,  and  the  other  3 
times  as  much  silver  as  copper.  How  much  must  be  taken 
from  each  to  weigh  a  kilogram,  of  which  the  silver  and  the 
copper  shall  be  equal  in  weight  ? 

41.  A  person  rows  a  distance  of  20  miles,  and  back  again, 
in  10  hours,  the  stream  flowing  uniformly  in  the  same 
direction  all  the  time ;  and  he  finds  that  he  can  row  2  miles 
against  the  stream  in  the  same  time  that  he  rows  3  miles 
with  it.    Find  the  time  of  his  going  and  returning. 


228  ELEMENTS  OF  ALGEBRA 

42.  A  and  B  can  do  a  piece  of  work  in  m  days,  A  and  C 
can  do  the  same  piece  in  n  days,  and  B  and  C  can  do  it  in 
p  days.     Find  in  how  many  days  each  can  do  the  work. 

43.  For  $26.25  we  can  buy  either  32  pounds  of  tea  and 
15  pounds  of  coffee,  or  36  pounds  of  tea  and  9  pounds  of 
coffee.     Find  the  price  of  a  pound  of  each. 

44.  A  pound  of  tea  and  3  pounds  of  sugar  cost  $1.50; 
but  if  sugar  were  to  rise  50  per  cent,  and  tea  10  per  cent, 
they  would  cost  $  1.75.     Find  the  price  of  tea  and  sugar. 

45.  A  person  possesses  a  certain  capital  which  is  invested 
at  a  certain  rate  per  cent.  A  second  person  has  $5000 
more  capital  than  the  first  person,  and  invests  it  at  1  per  cent 
more;  thus  his  income  exceeds  that  of  the  first  person  by 
$400.  A  third  person  has  $7500  more  capital  than  the 
first,  and  invests  it  at  2  per  cent  more;  thus  his  income 
exceeds  that  of  the  first  person  by  $  750.  Find  the  capital 
of  each  person  and  the  rate  at  which  it  is  invested. 

46.  Two  plugs  are  opened  in  the  bottom  of  a  cistern  con- 
taining 192  gallons  of  water ;  after  3  hours  one  of  the  plugs 
becomes  stopped,  and  the  cistern  is  emptied  by  the  other  in 
11  more  hours ;  had  6  hours  occurred  before  the  stoppage, 
it  would  have  required  only  6  hours  more  to  empty  the 
cistern.  How  many  gallons  will  each  plug-hole  discharge 
in  an  hour,  supposing  the  discharge  uniform  ? 

47.  A  certain  number  of  persons  were  divided  into  3 
classes,  such  that  the  majority  of  the  first  and  second  classes 
together  over  the  third  was  10  less  than  4  times  the  majority 
of  the  second  and  third  together  over  the  first ;  but  if  the 
first  class  had  30  more,  and  the  second  and  third  together 
29  less,  the  first  would  have  outnumbered  the  last  2  classes 
by  1.  Find  the  number  in  each  class  when  the  whole  num- 
ber was  34  more  than  8  times  the  majority  of  the  third  class 
over  the  second. 


PROBLEMS  229 

48.  Two  persons,  A  and  B,  could  finish  a  work  in  7n 
days ;  they  worked  together  n  days,  when  A  was  called  off, 
and  B  finished  it  in  p  days.    In  what  time  could  each  do  it  ? 

49.  The  fore-wheel  of  a  carriage  makes  6  revolutions 
more  than  the  hind-wheel  in  going  120  yards ;  if  the  circum- 
ference of  the  fore-wheel  be  increased  by  \  of  its  present 
size,  and  the  circumference  of  the  hind-wheel  by  \  of  its 
present  size,  the  6  will  be  changed  to  4.  Required  the  cir- 
cumference of  each  wheel. 


CHAPTER  XVI 
EVOLUTION.     IRRATIONAL  NUMBERS 

213.  An  nth  root  of  a  given  number  is  a  number  whose 
nth  power  is  equal  to  the  given  number. 

U.g.,  one  second  root  of  4  is  2,  since  2^  =  4. 
Another  second  root  of  4  is  —  2,  since  (—  2)2  =  4. 
A  third  root  of  —  8  is  —  2,  since  (—  2)^  =  —  8. 

A  second  root  of  a  number  is  usually  called  a  square  root ; 
and  a  f^irc?  root  a  c«6e  root. 

214.  The  radical  sign,  ^,  written  before  a  number,  denotes 
a  root  of  that  number. 

The  radicand  is  the  number  Avhose  root  is  required. 

The  index  is  the  number  which,  written  before  and  a  little 
above  the  radical  sign,  indicates  ivhat  root  is  required. 
When  no  index  is  written,  2  is  understood. 

E.g.,  -y/lG  or  ^16  denotes  a  second,  or  square,  root  of  16 ; 
16  is  the  radicand,  and  2  is  the  index. 

The  expression  ^u  denotes  an  nth  root  of  u ;  u  is  the 
radicand,  and  n  the  index. 

215.  Since  by  definition  {-^uY  =  u,  it  follows  that  ^u 
is  one  of  the  n  equal  factors  of  u. 

216.  A  rational,  or  commensurable,  number  is  any  whole  or 
fractional  number. 

A  rational  expression  is  one  which  ca7i  be  written  without 
using  an  indicated  root.  All  the  expressions  in  the  previous 
chapters  are  rational  expressions. 

230 


EVOLUTION  231 

217.  A  perfect  nth.  power  is  a  number  or  expression  whose 
»ith  root  is  a  rational  number  or  expression. 

E.g.,  since  ^'25  =  5,  25  is  2i perfect  square. 

Since  V—  Sa;^^  =  —  2  icy^,  _  8  ic^?/^  is  a  perfect  cube. 

Prior  to  §  238  each  radieand  will  be  a  perfect  power  of  a 
degree  equal  to  the  index  of  the  root. 

218.  Two  roots  are  said  to  be  like  or  unlike  according  as 
their  indices  are  equal  or  unequal. 

An  even  root  is  one  whose  index  is  even;  as,  -^a?. 
An  odd  root  is  one  whose  index  is  odd;  as,  -y/27. 

219.  Number  of  roots. 

(i)  An  arithmetic  niuiiber  has  one,  and  only  one,  nth  root. 

Any  odd  power  of  a  positive  or  negative  base  has  the 
same  quality  as  the  base  itself;  hence, 

(ii)  A  positive  or  a  negative  number  has  one  odd  root  of  the 
same  quality  as  the  number  itself. 

E.g.,  one  value  of  \/+ 27  is  +  S,  since  (+  3)3  =  +  27. 
Again,  one  value  of  v^—  32  is  —  2,  since  (—  2)^  =  —  32. 

If  two  numbers,  opposite  in  quality,  are  arithmetically 
equal,  their  like  even  powers  are  the  same  positive  number ; 
hence, 

(iii)  A  jyositive  number  has  two  even  roots,  which  are  arith- 
metically equal,  and  opjwsite  in  quality. 

E.g.,  two  values  of  V+81  are  +  9  and  -  9,  since  (+  9)2  or  (-  9)'^ 
is  +  81. 

Again,  two  values  of  \/+  81  are  +3  and  —3,  since  (+3)*  or 
(_3)Ms  +81. 

Any  even  power  of  a  positive  or  a  negative  number  is 
positive;  hence  an  even  root  of  a  negative  number  cannot  be 
a  positive  or  a  negative  number. 

Even  roots  of  negative  numbers  give  rise  to  new  quality- 
numbers,,  which  will  be  considered  in  Chapter  XVIII. 


232  ELEMENTS   OF  ALGEBRA 

220.  The  principal  root  of  a  positive  number  is  its  positive 
root. 

The  principal  odd  root  of  a  yiegative  number  is  its  negative 
root.  E.g.,  +  4  is  the  principal  square  root  of  16,  and  —  3 
is  the  principal  cube  root  of  —  27. 

Unless  the  contrary  is  stated,  the  radical  sign  will  hereafter 
be  understood  as  denoting  only  the  principal  root. 

221.  The  like  principal  roots  of  equal  numbers  are  equal ; 
hence, 

The  like  principal  roots  of  identical  expressions  are  identical 
expressions. 

222.  Evolution  is  the  operation  of  finding  any  required 
root  of  a  number  or  expression. 

In  the  statement  of  the  following  principles  of  joots,  by 
"  the  root "  is  meant  "  the  principal  root." 

223.  The  exponent  of  any  base  in  the  root  is  equal  to  the 
exponent  of  that  base  in  the  radicand  divided  by  the  index  of 
the  root;  and  coyiversely. 

That  is,  ^a"'"  =  a'".  (1) 

Proof     By  §  118,       {aTf  =  a"*". 

Hence,  by  ?  221,  a"*  =  -^a"*",  and  conversely  (1). 

E.g.,  ^a^  =  a^^^  =  a^ ;  ^x^^  =  x^. 

224.  The  nth  root  of  a  product  is  equal  to  the  product  of  the 
nth  roots  of  its  factors  ;  and  conversely. 

That  is,  V~^b=ya'^b.  (1) 

Proof    By  §  119,  {^a  •  ^by  =(^ay(-^by=ab. 

Hence,  by  §  221,        -^a  •  -y/b  =  Vab,  and  conversely  (1). 


Ex.  1.    v/-  32  aio  =  V-  32  •  ^a^^  =  -2a^. 


Ex.  2.    V-  aPh^  =  </^-\  ■  ^«9  .  ^66  =  _  a%\ 


EVOLUTION  233 

Observe  that,  from  this  principle,  it  follows  that  the  ?ith 
root  of  any  real  number  is  the  nth  root  of  its  quality-unit 
into  the  nth  root  of  its  arithmetic  value.     Thus, 

^/■+27=^Ti-^27  =  +  3;  </^^^  =  V^^ -  ^S2  =  -2. 

225.  The  nth  root  of  the  quotient  of  two  numbers  is  equal  to 
the  quotient  of  their  nth  roots;  and  conversely. 

That  is,  Va/b  =  ^a/^b.  (1) 

Proof     By  §186,        f^X ^^^^ 

Hence,  by  §  221,        ^a/^h  =  Va^,  and  conversely  (1). 

Ex.1.  <p^^^=J^E3ggg  §225 

^  216  ai2        ^216^ 

=  (-5x2)7(6  a*).  §224 

Ex.  2.  ^/TiJy^j^Eiggg  §§  167,  225 

=  -2xyV(a62;58).  §224 


Exercise  86. 
Reduce  to  a  rational  form  the  following  expressions : 

1.  V4^*.  8.    ^-64a^/.        14     ^ /400cW^ 

2.  V9a^/.  9.    ^343  a^'b\  ,1125 a'b^ 

3.  V25W«.  10.    .^Sr^^  ''*    V216^ 


'■f 


12.    ^/32^«.  1^-    \_aW' 

6.    Sy-8a^/. 


7.    V-aW^. 


...  vf •      "■  ^" 


128 


234  ELEMENTS   OF  ALGEBRA 


19.    J?-l  21.    J-  +  ^-  23.    J5-1. 

^3     9  >i5     25  Ve     18 


-•  Vf-i-     -•  \fl-     -•  \ 


226.    77ie  s^7i  roo^  of  the  rth  power  of  a  number  is  equal  to 
the  rth  power  of  its  sfh  root ;  and  conversely. 


That  is, 

^'a^^{-^ay. 

(1) 

Proof     Let 

</a  =  B', 

(2) 

then, 

a  =  B\ 

§128 

.'.  a^={By  =  B-, 

§§  128, 118 

Hence,  by  221, 

</a^=B\ 

(3) 

From  (2), 

{^ay  =  B\ 

(4) 

From  (3)  and  (4),  by  §  32,  we  obtain  (1). 


Ex.  1.    \/(64/125)-«  =  (V64/125)2 

=  (4/5)2  =  16/25. 


Ex.2.    V(81x2«c*)3=(>/8r^«c*)3 

=  (9x«c2)3  =  729x3«c6. 

227.  The  sth  root  of  the  qth  root  of  a  nmnber  is  equal  to 
the  qsth  root  of  the  number ;  that  is, 

Proof  If  a  number  is  resolved  into  q  equal  factors,  and 
then  each  one  of  these  q  equal  factors  is  resolved  into  s 
equal  factors,  the  number  will  be  resolved  into  qs  equal 
factors  ;  that  is,  ,  /  y^  _  gy^ 

Ex.  1.  ^ v(2^  ^^y^'^)  =  \/(2®  ^y^^) = 2  xy^-  §  227 

Ex.  2.    ^  V(25-^  X  93)  =  </(56  X  36)  =  5  X  3  =  15. 


EVOLUTION  235 


Exercise  87. 

Reduce  each  of  the  following  expressions  to  a  rational 
form : 

1.  ^^{64.  aV).  ^      sW'\ 

2.  V^'(27'x64^.  \a«"ft«« 

3.  V^(aWa-).  ''    ^V(729a«a-). 

4.  Vae/W-  '^-    ^(27/64)^. 

6.  <WW^'-  ''•  ^(^^^"^^)- 

\.064a«'  14.    ^(8  aV?/»)*. 

228.  Square  root  by  inspection.  When  a  perfect  square 
can  be  factored  by  inspection,  its  square  root  is  found  by 
inspection. 

Ex.  1.  30  rt»  +  ft*  -  12  a2fta  =  (6  a^  -  b^y-,  or  (ft2  -  6  a2)2. 

.-.  V(36  a»  +  ft*  -  12  a2ft2)  =  6  a^  -  ft^,  or  ft2  -  6  a^. 

Ex.-  2.  Find  the  square  root  of  the  first  eight  expressions  in  each 
of  the  exercises  52  and  53. 

229.  To  show  how  to  find  the  square  root  of  any  perfect 
square,  we  must  show  how  to  reverse  the  process  of  squanng 
any  expression. 

J5r.gr.,  squaring  expression  (1)  we  obtain  expression  (2). 

x8  +  rx2  +  sx  (1) 

xfi  +  2.ric6  +  (r2  +  2  s)a;*  +  2  rsx^  +  sH^  (2) 

Hence  if  (2)  is  taken  as  a  radicand,  (1)  is  its  square  root. 

Now,  the  square  root  of  the  Jirst  term  of  the  radicand  (2)  is  the 
first  term  of  the  root  (1). 

If  we  subtract  from  (2)  the  square  of  the  ^rs«  term  of  (1),  the^rsi 
term  of  the  remainder  is  2  ra^.  Dividing  2  rx^  by  twice  the  first  term 
of  the  root,  2  x^,  we  obtain  rx2,  the  second  term  of  the  root. 


236  ELEMENTS   OF  ALGEBRA 

If  we  subtract  from  the  radicand  the  square  of  the  sum  of  the  Jirst 
two  terms  of  the  root,  (x^  +  rx^)^,  the  first  term  of  the  remainder 
is  2  sx^.  Dividing  2  sx*  by  twice  the  first  term  of  the  root,  we  obtain 
sx,  the  third  term  of  the  root. 

This  example  illustrates  the  following  principles  (i)  and  (ii). 

If  the  terms  of  a  perfect  square  and  its  square  root  are 
arranged  in  descending  (or  ascending)  powers  of  some  letter, 

(i)  TJie  square  root  of  the  first  term  of  the  radicand  is  the 
first  term  of  the  square  root.  , 

(ii)  If  the  square  of  the  first  term  in  the  root,  or  the  square 
of  the  sum  of  its  first  two  or  more  terms  is  subtracted  from  the 
radicand,  and  the  first  term  of  the  remainder  is  divided  by 
twice  the  first  term  of  the  root,  the  quotient  will  be  the  next 
term  of  the  root. 

Proof  Let  ^  stand  for  any  number  of  terms  of  the 
square  root  of  any  perfect  square,  and  B  for  the  rest ;  the 
terms  of  A  and  B  being  arranged  in  descending  (or  ascend- 
ing) powers  of  the  same  letter,  and  every  term  of  A  being 
of  a  higher  (or  lower)  degree  than  any  term  of  B. 

By  §  120,  we  have  the  identity 

A'  +  2AB-\-B'  =  (A  +  By.  .     (1) 

Let  A  denote  only  the  fij^st  term  of  the  root ;  then,  since 
-^A^  =A,  we  have  (i). 

Let  A  denote  the  first  one  or  more  terms  of  the  root; 
then,  if  we  subtract  A^  from  the  radicand,  the  remainder  is 
2  AB  4-  J5^.  Let  a  denote  the  first  term  of  A,  and  5  the  first 
term  of  B ;  then,  supposing  the  remainder  2  AB  -f-  B^  to  be 
arranged  in  descending  (or  ascending)  powers  of  the  letter 
of  arrangement,  2ab  will  be  its  first  term.  Hence,  as 
2  ab  -i-  2  a  =  b,  we  have  (ii). 

E.g.,  by  (i),  the  first  term  of  the  square  root  of 

16  x^  -  24  yx^  +  25  ?/%2  _  12  y^x  +  4:y^  (3) 

is  \/l6  X*,  or  4  jr2 ;  and  by  (ii)  the  second  term  is  -  24  yx^  -7-2(4  x^), 
or  —  Zyx. 


EVOLUTION  237 

The  radicand  (3)  less  (ix^-S  yxY  is  16  y'^x'^  -  12  t/^x  +  4  y2. 
Hence,  by  (ii),  the  next  term  of  the  root  is  16  y'^j^^  -^  2(4  x^),  or  2/2. 
The  radicand  (3)  less  (4  x2  -  3  ?/x  +  2  y2)2  jg  zero. 
Hence,  the  square  root  of  (3)  is  4  x2  —  3  yx  +  2  y2. 

Instead  of  finding  each  square  independently,  some  labor 
can  be  saved  by  using  the  relation 

A'  +  (2A  +  h)h  =  {A  +  h)\  (2) 

and  thus  making  use  of  the  previous  square.  Thus  the 
work  in  the  example  above  is  usually  written  as  below : 

16  X*  -  24  yx^  +  25  ^2^2  -\2yH-\-^  y*(4  x2  -  3  i/x  +  2  2/2. 

16  X* 

8  x2  -  3  yx)  -  24  yx^ 

-  24  yx^  +    9  y2x2 

8x2-6yx    +22/2      )      162/2x2 

16  ygx2  -  12  y^x  +  4  y^ 

Subtracting  from  the  radicand  the  square  of  the  first  term  of  the 
root,  (4  x2)2,  the  first  term  of  the  remainder  is  -  24  yx'. 

By  (ii),  the  second  term  of  the  root  is  -  24  yx'  -^  2(4  x2),  or  -  3  yx. 

Write  2(4x2)-3yx  to  the  left  of  the  first  remainder,  multiply  it 
by  -  3  yx,  and  subtract  the  product  from  the  first  remainder. 

Then,  by  (2),  we  have  subtracted  in  all 

(4  x2)2  +  (2  .  4  x2  -  3  yx) ( -  3  yx),  or  (4  x2  -  3  yx)2. 

By  (ii),  the  next  terra  of  the  root  is  16  y2x2  -4-2(4  x2),  or  2  y2. 

Write  2(4x2 -3yx)+2y2  to  the  left  of  the  second  remainder, 
multiply  it  by  2  y2,  and  subtract  the  product  from  the  second  remain- 
der.    Then,  by  (2),  we  have  subtracted  in  all 

(4  x2  -  3  yx)2  +  (8  x2  -  6  ?yx  +  2  y2)2  y2,  or  (4  x*  -  3  yx  -f  2  y2)2. 

As  there  is  no  remainder,  the  required  root  is  4  x2  —  3  yx  +  2  y2. 
Observe  that  we  could  just  as  well  write  radicand  (3)  in  ascending 
powers  of  x,  or  what  is  the  same  thing,  begin  with  its  last  term. 

16x*-24yx8-f  25y2x2-12y8x  +  4y*.  (3) 

Thus,  by  (i),  the  last  terra  of  the  square  root  of  (3)  is  \/4y2,  or  2  y2 ; 
and,  by  (ii),  the  term  before  the  last  is  -  12  y'x  -r-  2  (2  y2),  or  -  3  yx, 
which  agrees  with  the  result  above. 


238  ELEMENTS   OF  ALGEBRA 

Ex.    Find  the  square  root  of  4  x*  —  8  r^  +  4  x  +  1.  (2) 

The  Jirst  term  of  the  root  is  2  x'^,  and  the  second  term  is  —  8  x^-f-4  x^, 
or  —  2x. 

The  last  term  is  ]  or  —  1.  If  the  last  term  is  —  1,  the  term  before 
the  last  is  4  a;  H-(—  2),  or  —  2  x,  which  is  the  second  term  as  found 
above. 

But  (2  ic2  —  2  a;  —  1)^  =  the  given  expression  ; 

hence  2  x^  _  2  x  —  1  is  the  required  root. 

If  we  took  —  2  ic^  as  the  first  term  of  the  root,  the  second  term 
would  be  2  x,  and  the  last  term  1. 

Note.  In  the  following  exercise  the  pupil  should  write  out  the 
root  at  once  by  (i)  and  (ii),  as  in  the  example  above  ;  but  he  should 
be  drilled  also  in  arranging  the  work  of  finding  and  subtracting  the 
successive  squares  as  on  page  237. 

Exercise  88. 
Find  the  square  root  of  the  following  expressions : 

1.  .T*  +  2it'3  +  3a^  +  2a;-f  1. 

2.  4:X*  —  Sa:^-\-4:X-\-l. 

3.  9x'-S6x^-^72x  +  3(J. 
'4.    4x^  +  4:X^  —  ^x-\-^Q. 

5.  x*-\-2x^y-{-3xY-^2xf  +  y*. 

6.  x*-2x^-{-^x'^--lx-\-j\. 

7.  16-9Gx  +  216x'-216x^-\-Slx\ 

8.  l-\-4.x-{-10x'-^12x^-\-9x\ 


,.    4.^4_4^_,_3^2_^_^ 


10.  l-xy-~\^-xPf  +  2a^y^-\-4.xY. 

11.  x^-4:X^-{-6x*-Sx^  +  9x^-4:X-\-4:. 

12.  9x^-12o(^  +  22x*-\-x^-{-12x-^4. 

13.  ct^-22a.'4  4-34a^4-121a^-374a;  +  289. 

14.  a^  —  ax-^^x^-{-Sa  —  4:X  +  16. 


EVOLUTION  239 

16.  a^  +  2a;^-f a^-4ar'-12a;*-8a^4-4a^  +  16a;-fl6. 

16.  (l-^2x'y-4.x(l-x)(l  +  2x). 

17.  x' -\-2a^(y  +  z)-{-x'(y' -hz'' -^^yz)  -{-2xyz(y  +  z)  +  fz\ 

18.  a^-2x  +  ^-{-\^x''-6a^. 

19.  -Sa^  +  ^  +  a*-5a-\-{ia\ 

20.  ia;*  +  4:x2  +  ^aar^  +  i«'-2a^-Jax. 

21.  24  +  16|!!_8.^^     32,. 

ar        y      y        X 
In  the  polynomial 

x3  +  X-^  +  X+l+-  +  -\  +  ^  (1) 

each  term  after  the  first  is  obtained  by  dividing  the  preceding  term 
by  x;  hence  we  regard  all  the  terms  in  expression  (1)  as  arranged 
according  to  the  powers  of  x. 

Arranging  the  given  expression  according  to  the  powers  of  x,  we 

have 

x^     8x     o^     S2y     my\ 
y^      y  X        x2 


4      a;      ar* 

-'-i 

9a2     6a     101 
a^       5  a;      25 

4.x      4ar' 
15a     9a* 

23 

'     x"       bx      Zb       1 

24.    4a.-*  +  32ar^H-96+^  +  i^. 
«*       ar 

230.  Cube  root  by  inspection.     When  a  perfect  cube  can  be 
factored  by  inspection,  its  cube  root  is  found  by  inspection. 

Ex.  1.    27  a^  -  54  a%  +  36  a^b'^  -  8  a%^  =  (3  ^2  _  2  ahy. 

.-.  ^(27  a«  -  54  a^h  +  36  a»62  _  8  a%^)  =  3  a^  -  2  aft. 

Ex.  2.    Find  the  cube  root  of  the  first  nine  expressions  in  exer- 
cise 60. 

231.  Cube  root  of  any  perfect  cube.     Let  A  stand  for  any 
number  of  terms  in  the  cube  root  of  any  perfect  cube  and 


240  ELEMENTS   OF  ALGEBRA 

B  for  the  rest;  the  terms  of  A  and  B  being  arranged  in 
descending  (or  ascending)  powers  of  the  same  letter,  and 
every  term  in  A  being  of  a  higher  (or  a  lower)  degree  than 
any  term  in  B. 

By  §  124  we  have  the  identity 

^3  +  3  A^B  4-  3  AB^  +  l^  =  {A-\-  Bf.  (1) 

(i)  Let  A  denote  the  first  term  of  the  root ;  then  from  (1) 
it  follows  that  the  cube  root  of  the  first  tenn  of  the  radicand 
is  the  first  term  of  the  root. 

(ii)  Let  A  denote  the  first  one  or  7nore  terms  of  the  root ; 
then  if  we  subtract  A^  from  the  radicand  the  remainder  is 
3A^B  +  SAB^  +  B\  Let  a  denote  the  first  term  of  A, 
and  b  the  first  term  of  B;  then  supposing  the  remainder 
3  A^B  +  3  AB"^  +  B^  to  be  arranged  in  descending  (or  ascend- 
ing) powers  of  the  letter  of  arrangement,  3a^b  will  be  its 
first  term.     But  3  a^6  -;-  3  a-  =  6 ;  hence. 

If  the  cube  of  the  first  term  of  the  root,  or  the  cube  of  the 
sum  of  its  first  two  or  more  terms,  is  subtracted  from  the 
radicand,  and  the  first  term  of  the  remainder  is  divided  by 
three  times  the  square  of  the  first  term  of  the  root,  the  quotient 
will  be  the  next  term  of  the  root. 

E.g.,  by  (i),  the^rs^  term  of  the  cube  root  of 

8  a;6  -  36  x5  +  66  a;4  -  63  ccS  +  33  x2  -  9  X  +  1 

is  v^8x6,  or  2 ^2  ;  and  by  (ii)  the  second  term  is  -  36 x^  ^  3 (2  x2)2, 
or  —  3  X. 

The  radicand  less  (2x'^-Sxy  is  12  x*  -  S6x^ +  Sox'^-9x  +  1. 
Hence,  by  (ii),  the  next  term  of  the  root  is  12  x*  -^  3  (2  x^)^,  or  1. 
The  radicand  less  (2  x^  —  3  x  +  1)*  is  zero. 
Hence  the  cube  root  is  2  x^  —  3  a;  +  1. 

Instead  of  finding  each  cube  independently,  some  labor 
can  be  saved  by  using  the  relation 

A'  -f  (3  A'  +  3^16  +  b')b  =  (A  +  by, 


EVOLUTION  241 

and  thus  making  use  of  the  previous  cubes.     Thus  the  work 
in  the  example  above  is  usually  written  as  below : 

I  2a;2-3a;+l 
Sofi-S6x^+6Qx^-6Sx^+3Sx^-9x-\-l 
08=  Sofi 

3a2=12x* 


3a6  +  62=         -18x3+  9x2 
3.42^12x4-30x3+27x2 


-36x5 
-36x5+54x*-27x» 


12x*-36x3 
12x4-36x8-33aj2-9a;  +  l 


3^&  +  &2=  6x2-9x+l 

By  (i),  the  first  term  of  the  root  is  y/8afi,  i.e.,  a  =  2x^. 

Subtract  (2x2)8;  then,  by  (ii),  the  second  term  of  the  root  is 
-36x5-3(2x2)2,  i.e.,  6  =  -3x. 

Hence  3a2  +  3a6  +  52  =  12  x*  -  ISx^  +  9x2. 

Multiply  this  sum  by  —  3  x,  and  subtract  the  product  from  the  first 
remainder.  Then  in  all  we  have  subtracted  a*  +  (3  a2  +  3  aft  +  b^)b, 
or  (a  +  6)3 ;  that  is,  we  have  subtracted  (2  x2  —  3  x)*,  since  a  =  2  x^ 
and  6  =  —  3  X. 

Let  A  =  the  terms  of  the  root  already  found  =  2  x2  —  3x, 
and  b  =  the  next  term  of  the  root  =  12  x*  -=-  3(2  x2)2  =  1 ; 
then  3^2  4.  3^5  4.  52  =  12  X*  -  36x«  +  33x2  -  9 x  +  1. 

Multiply  this  sum  by  1,  and  subtract  the  product  from  the  second 
remainder.     Then  in  all  we  have  subtracted 

(A  +  6)8,  or  (2  x2  -  3  X  +  I)'. 

As  there  is  no  remainder,  the  required  root  is  2  x2  —  3  x  +  1. 

We  could  just  as  well  write  the  radicand  in  ascending  powers  of  x, 
or,  what  is  the  same  thing,  begin  with  the  last  term. 

8x«-36x5  +  66x*-63x8  +  33x2-9x  +  l.  (1) 

Thus,  by  (i),  the  last  term  of  the  cube  root  of  (1)  is  -^1,  or  1 ;  and, 
by  (ii),  the  term  before  the  last  is  —  9x  -^  3  •  I2,  or  -  3x,  which  agrees 
with  the  result  above, 

Ex.   Find  the  cube  root  of 

27 +  108X  + 90x2 -80x3-  60x*  +  48x5  -  8x«.  (1) 

The  first  term  is  3,  and  the  second  is  108  x  -f-  3  •  3*,  or  4  x. 

The  last  term  is  —  2  x2,  and  the  term  before  the  last  is 
48x5 -^3(- 2x2)2,  or  4x. 

Since  (3  +  4  x  —  2  x2)8  =  the  given  expression  ; 

3  +  4x  —  2x2  =  the  required  root. 


242  ELEMENTS   OF  ALGEBRA 

Exercise  89. 
Find  the  cube  root  of  the  following  expressions : 

1.  l-{.3x-^6x'-^7x'-{-Qx'  +  3x^-{-x^ 

2.  f  -3f  -\-6y'  -  7  f  ^  6f  -8y  +  1. 

3.  l-6x  +  21i^-Uiif-{-()3x'-o4:X^  +  27x^. 

4.  8  a«  -  36  a^  +  (^(^  a'  -  63  a^  +  33  a^  -  9  a  +  1. 

5.  Sx^-{-12x^-30x*-3ox^-^4.ox^-{-27x-27. 

6.  27x^-27x^-lSx'  +  17x^  +  6s^-3x-l. 

7.  24  xY  +  96  xY  -  6  :t'V  H"  a^'  -  9^  a.y  +  64  /  -  56  a^/. 

8.  27  a;«  _  54  a^^a  +  117  a;V  -  116  x'a^  +  117  a^-a*  -  54  xa' 
+  27  a«. 

9.  216  +  342  x-  +  171  X*  +  27  x^-27^-  109  a.-^  -  108  a;. 

10.  a:3_9^^27_27 

X       x^ 

11.  --6a;^  +  12ar2/'-8/. 

12.  ^V  — +  — -4-^  +  — --• 
l^      y'        y  X       x?      x^ 

13. 


14. 


15.    — i , ^  +  -^  + J7  +  — . 

232.  Higher  roots.  The  fourth,  fifth,  or  any  other  root  of 
a  perfect  power  can  be  obtained  by  a  method  based  on  one 
of  the  following  identities: 

A'  -{■4:A'B  +  6^2j52  +  4.AE'-\'B*={A  +  By,       (1) 

A''{-5A'B-^10A'B'+10A'B'-\-5AB'+B'^iA+By.     (2) 


x' 

27 

|42.- 

-7  + 

18_ 

X 

27      27 
a^^'^ar'* 

a 

/^«      7  +  ^' 
6              6^ 

3  a' 
b' 

352       ?>3 

GOa;^ 
2/^ 

80^3 
2/^ 

90  ar^ 
2/^ 

+  8a 

f      108  a; 

y 

EVOLUTION  243 

If  the  terms  of  a  perfect  fourth  power  are  arranged  in 
descending  (or  ascending)  powers  of  some  letter,  from  (1) 
it  follows  that  the  first  term  of  the  root  is  the  fourth  root 
of  the  first  term  of  the  radicand ;  that  the  second  term  of 
the  root  is  the  second  term  of  the  radicand  divided  by  four 
times  the  cube  of  the  fii'st  term  of  the  root ;  and  that  the 
last  term  of  the  root  is  the  fourth  root  of  the  last  term 
of  the  radicand.     Similarly  for  any  other  higher  root, 

E.g.^  the  first  term  of  the  fourth  root  of 

81a;*+ 108x3+54x2  +  12a;  +  l  (1) 

is  v^81  X*,  or  3ic  ;  the  second  term  is  108  x^  ^  4(3x)8,  or  1,  which  we 
know  to  be  the  last  term  of  the  root. 

Since  (3  a; +1)*=  the  radicand  (1);  3ic+l  is  the  fourth  root 
of  (1). 

Again  the  first  term  of  the  fifth  root  of 

32x6- 80x*  + 80x8 -40x2+ lOx-1  ^        (2) 

is  \/32x^,  or  2  X ;  the  second  term  is  -  80x*  h-  5(2 x)*,  or  —  1,  which 
we  know  to  be  the  last  term  of  the  root. 

Since  (2x  -  1)5  =  the  radicand  (2);  2x  -  1  is  the  fifth  root  of  (2). 

The  fourth  root  can  also  be  obtained  by  finding  the  square 
root  of  the  square  root;  and  the  sixth  root,  by  finding  the 
cube  root  of  the  square  root.  Similarly  for  any  other  root 
whose  index  is  not  a  prime  number. 

Exercise  00. 
By  inspection  find  the  fourth  root  of  the  expressions : 

1.  16  a*  -  96  a»aj  +  216  aV- 216  aa^  + 81  a;*. 

2.  x^-%a?a  +  24.a?a'-^2xa^  +  Ua\ 

3.  1  +  4  a  4-  4  a^-f- 10  a«+  a«+  10  a^-\- 16  a^-f  16  a*+  19  a\ 

By  inspection  find  the  fifth  root  of  the  expressions : 

4.  80  a^a^  -  80  ax^  +  32  a.-*  -  40  a^x  -  a^  +  10  a^x. 

6.   90aV~16aa^  +  a^-270aV  +  405a*aj-243a«. 


244  ELEMENTS   OF  ALGEBRA 

By  inspection  find  the  sixth  root  of  the  expressions : 

6.  192  a;  +  64  +  240  a^^  _^  ^.e  _^  i2a^  +  60  a;^  +  160  a:^. 

7.  1215  a*  -  1458  a'  -  540  a^  +  135  a^  -  18  a  +  1  +  729  a«. 

8.  60  a'x'-  16  a^x^-{-  64  a^-^  x^-  12  aa^+  240  aV-  192  a'x. 

ROOTS   OF   DECIMAL  NUMBERS. 

233.  Square  roots.  ^1  =  1,  and  ^10^  =  10 ;  hence  the 
square  root  of  any  number  between  1  and  100  lies  between 
1  and  10 ;  that  is,  if  a  number  contains  one  or  two  integral 
figures,  its  square  root  contains  one  integral  figure. 

Again,  ^100  =  10,  and  V^O^OO  =  100 ;  hence  the  square 
root  of  any  number  between  100  and  10000  lies  between  10 
and  100 ;  that  is,  if  a  number  contains  three  or  four  integral 
figures,  its  square  root  contains  two  figures ;  and  so  on. 

Hence,  in  finding  the  square  root  of  a  decimal  number, 
the  first  step  is  to  divide  its  integral  figures  into  groups  of 
two  figures  each,  beginning  at  units'  place. 

We  thus  determine  the  number  of  integral  figures  in  the 
root,  and  indicate  the  part  of  the  number  from  which  each 
figure  of  the  root  is  to  be  obtained. 

The  group  to  the  left  may  contain  only  one  figure. 

E.g.,  in  the  square  root  of  5  38  24  there  are  hundreds,  tens,  and 
units  ;  and  the  hundreds'  figure  is  the  square  root  of  the  greatest  per- 
fect square  in  5  ;  that  is,  the  hundreds'  figure  is  2. 

234.  From  §  233,  we  have  the  following  principle : 

(i)  The  first  figure  in  the  square  root  of  a  decimal  number 
is  the  square  root  of  the  greatest  perfect  square  in  the  first,  or 
left-hand,  group  of  figures. 

Let  A  stand  for  the  number  denoted  by  one  or  more  of 
the  first  figures  of  the  root,  and  B  stand  for  the  number 
denoted  by  the  rest ;  then  the  root  is  A-}- B. 

Hence  the  radicand  =  A^-\-(2A  +  B)B.  (1) 

Subtract  A^  and  then  divide  by  2  ^  +  J5, 

(radicand  -A^^{2A-^B)=B.  (2) 


EVOLUTION  245 

Let  h  stand  for  the  number  denoted  by  the  first  figure  in 
B,  i.e.,  the  figure  whose  order  of  units  is  the  next  lower  to 
the  lowest  in  A ;  then  from  (2) 

(radicand  -A^-i-2A>b.  (3) 

From  inequality  (3),  we  have  the  following  principle : 

(ii)  If  the  square  of  the  first  part  of  the  root  is  subtracted 
from  the  radicand,  and  the  remainder  is  divided  by  twice  this 
part  of  the  root,  the  quotient  will  be  greater  than  the  next  figure 
of  the  root. 

E.g.,  by  (i),  W\Q  first,  or  hundreds,  figure  in  the  square  root  of 

5  47  56 

is  2  ;  since  4  is  the  greatest  perfect  square  in  5. 
The  radicand  less  (200)2  is  14756. 
Hence,  by  (ii),  the  tens  figure  of  the  root  cannot  exceed 

14756 -2(200),  or  3  tens. 

The  radicand  less  (230)2  is  1856  ;  hence  the  root  is  greater  than  230, 
and  3  is  the  tens  figure  of  the  root. 

By  (ii),  the  units  figure  of  the  root  cannot  exceed 

1856  H-  2(230),  or  4. 

The  radicand  less  (234)2  jg  zero. 
Hence  the  square  root  of  54756  is  234. 

Instead  of  finding  each  square  independently,  much  labor 
can  be  saved  by  using  the  relation  (§  229) 

A'-\-{2A-^b)b={A^by, 

and  thus  making  use  of  the  previous  square. 

The  work  in  the  example  above  is  usually  written  as 

below : 

5  47  56(234 
^2  =(200)2=  4  00  00 

2^  +  6  =  2(200)  +  30  =     430)14756 
(2  ^  +  6)&  =  430x30=  129  00 

2^  +  6=2(230)4-    4=       464)18  56 
(2^  +  6)6  =  464x    4=  18  56 


246  ELEMENTS   OF  ALGEBRA 

At  first  A  =  200.  Subtracting  A^,  or  200^,  from  the  radicand,  and 
dividing  the  remainder,  14756,  by  2  A,  or  2(200),  we  find  that  the  tens 
figure  of  the  root  cannot  exceed  3. 

Multiply  2  A  +  b,  or  430,  by  6,  or  30,  and  subtract  the  product ; 
then  in  all  we  have  subtracted  A- +  (2  A+b)b,  or  (A-\-by;  that  is,  230"'2. 

Now         let  A  =  230,  the  part  of  the  root  already  found, 
and  b  =  the  next  figure  of  the  root. 

Dividing  the  remainder  1856  by  2  A,  or  460,  we  find  that  the  units 
figure  of  the  root  cannot  exceed  4. 

Multiply  2  ^  +  6,  or  464,  by  b,  or  4,  and  subtract  the  product ;  then 
in  all  we  have  subtracted  A^  +  (2A  +  b)b,  or  (A  +  6)2  ;  that  is,  (234)-^. 
Omitting  the  ciphers  and  explanation,  and  in  each  remainder  writing 
the  next  group  of  figures  only,  the  work  will  stand  as  below  : 

5  47  56(234 

4 

43)147 
129 
464)18  56 
18  56 

235.  If  a  number  has  decimal  places,  its  square  will  have 
twice  as  many.     E.g.,  O.S^  =  0.64 ;  0.25^  =  0.0625. 

Hence  to  determine  how  many  decimal  figures  there  will 
be  in  the  square  root  of  a  number,  we  divide  its  decimal 
figures  into  groups  of  two  figures  each,  beginning  at  the 
decimal  point.  If  the  group  to  the  right  does  not  contain 
two  figures,  a  cipher  must  be  annexed. 

Ex.   Find  the  square  root  of  5727.2976. 
Formula,  A^ +(2  A  +  b)b  =  (A  +  by. 

75  27.29  76(86.76 

64 

166)1127 
9  96 
1727)13129 
120  89 
17346)10  40  76 
10  40  76 
Here  at  first  A  =  SO,  b  =  6  ;  next  ^  =  86,  6  =  0.7  ,  next  A  =  86.7, 
b  =  0.06. 


EVOLUTION 


247 


Exercise  01. 

Find  the  square 

!  root  of  the  numbers 

: 

1.  2916. 

9.  29376400. 

17. 

0.0022448644. 

2.  2601. 

10.  52.2729. 

18. 

0.68112009. 

3.  17956. 

11.  53.7289. 

19. 

25/49. 

4.  33489. 

12.  883.2784. 

20. 

64/81. 

5.  119025. 

13.  1.97262025. 

21. 

121/36. 

6.  15129. 

14.  3080.25. 

22. 

144/49. 

7.  103041. 

15.  41.2164. 

23. 

169/196. 

8.  835396. 

16.  384524.01. 

24. 

225/289. 

236.  Cube  root.  Since  ^1  =  1,  and  ^1000  =  10,  it  fol- 
lows that  the  cube  root  of  any  number  between  1  and  1000 
lies  between  1  and  10 ;  that  is,  if  a  number  contains  one, 
two,  or  three  integral  figures,  its  cube  root  contains  one 
integral  figure.  Again,  ^1000  =  10,  and  ^1000000  =  100; 
hence,  if  a  number  contains  four,  five,  or  six  integral  figures, 
its  cube  root  contains  two  integral  figures ;  and  so  on. 

Hence,  to  determine  how  many  integral  figures  there  are 
in  the  cube  root  of  a  number,  we  divide  its  integral  figures 
into  groups  of  three  figures  each,  beginning  at  units'  place. 
The  last  group  to  the  left  may  contain  only  one  or  two 
figures. 

When  the  figures  of  a  number  have  been  divided  into 
groups  of  three  figures  each,  from  what  precedes  it  follows 
that, 

(i)  The  first  figure  in  the  cube  root  of  a  decimal  number  is 
the  cube  root  of  the  greatest  cube  in  the  first,  or  left-hand^ 
grouj)  of  figures. 

Using  a  notation  analogous  to  that  in  §  234,  we  have 
radicand  -  A''  =  (3  A'  -^  3  AB  +  B^  B. 
'.  (radicand  -  .1'')  -  (3  A-  -\- 3  AB -^  B^  =  B. 
.-.  (radicand -yl^) --3. 42  >  6.  (1) 


248  ELEMENTS   OF  ALGEBRA 

From  inequality  (1)  it  follows  that, 

(ii)  If  the  cube  of  the  first  part  of  the  root  is  subtracted 
from  the  radicand  and  the  remainder  is  divided  by^  three  times 
the  square  of  this  part  of  the  root,  the  quotient  will  be  greater 
than  the  next  figure  of  the  root. 

E.g.,  by  (i),  the  Jirst  or  tens^  figure  in  the  cube  root  of 

614  125 

is  8,  since  8^,  or  512,  is  the  greatest  perfect  cube  in  614. 
The  radicand  less  (80)^  is  102125. 
Hence,  by  (ii) ,  the  units'  figure  of  the  root  cannot  exceed 

102125  --  3(80)2,  or  5. 

The  radicand  less  (85) ^  is  zero. 
Hence,  the  required  root  is  85. 

Instead  of  finding  each  cube  independently,  much  labor 
can  be  saved  by  using  the  relation 

A'  -{-  {S  A'  +  3  Ab  -\-  b')b  =  (A  +  by, 

and  thus  making  use  of  the  previous  cubes. 

Thus,  the  work  in  the  example  above  is  usually  written 
as  below,  without  the  explanations  to  the  left : 

614  125(85 

43  =  512  000 

3^2^3(80)2     =19200) 
3^6  =  3.80-5=    1200 

62  =  52  =       25 


20425 


102  125 


102  125 


237.  If  a  number  has  decimal  places,  its  cube  will  have 
three  times  as  many.  Thus  0.2^  =  0.008 ;  0.12^  =  0.001728. 
Hence,  to  determine  how  many  decimal  figures  there  will 
be  in  the  cube  root  of  a  number,  we  divide  its  decimal 
figures  into  groups  of  three  figures  each,  beginning  at  the 
decimal  point. 

If  the  group  to  the  right  does  not  contain  three  figures, 
ciphers  must  be  annexed. 


lEBATIONAL  NUMBERS 


249 


Ex.   Find  the  cube  root  of  129554.216. 
Formula,  A^  -\-(iS  A^  +  S  Ab  +  b'^)b  =  (A  +  by. 


129  554.216(50.6 
125 


750000 
9000 

36 

759036 


4  554  216 


4  554  216 


Here  at  first  ^  =  50,  6  =  0;   next  A  =  50.0,  b  =  0.6. 


Exercise  92. 
Find  the  cube  root  of  the  numbers : 


1.  74088. 

2.  15625. 

3.  32768. 

4.  110592. 

5.  262144. 

6.  1481544. 


7.  103.823. 

8.  884.736. 

9.  1953125. 

10.  7077888. 

11.  2.803221. 

12.  12.812904. 


13.  56.623104. 

14.  264.609288. 

15.  1076890625. 

16.  8/27. 

17.  64/125. 

18.  343/1728. 


INCOMMENSURABLE   ROOTS,  OR  IRRATIONAL  NUMBERS. 

238.  The  nth  power  of  a  whole  number  is  evidently  a 
whole  number  which  is  a  perfect  ?ith  power ;  and  the  ?ith 
power  of  a  fraction  (whose  numerator  and  denominator  are 
prime  to  each  other)  is  a  fraction  whose  numerator  and  de- 
nominator are  perfect  nth.  powers  prime  to  each  other. 

Hence,  it  follows  that 

(i)  The  nth  root  of  a  tvhole  mimber  which  is  not  the  nth 
poicer  of  another  whole  number  is  not  a  commensurable 
number. 

(ii)  The  nth  root  of  a  fraction  whose  numerator  and  ck^ 
nominator  {prime  to  each  other)  are  not  the  nth  powers  oj 
whole  numbers,  is  not  a  commensurable  number. 


250  ELEMENTS  OF  ALGEBRA 

E.g.,  as  2  is  not  the  square  of  any  whole  number,  y/2  is  not  a  com- 
mensurable number,  and  therefore  is  not  as  yet  included  in  our  number 
system.    The  same  is  true  of  ^3,  y/b,  ^1  •••. 

Again,  as  the  terms  of  the  fraction  2/3  are  prime  to  each  other  and 
are  not  the  squares  of  whole  numbers,  ^(2/3)  is  not  a  commensurable 
number. 

239.  To  enlarge  our  number  concept  so  as  to  give  mean- 
ing to  such  expressions  as  ^2,  ^5,  etc.,  we  assume  the 
identity 

to  hold  when  the  radicand  u  is  not  a  perfect  /ith  power. 

E.g..,  yJ2  is  the  number  whose  square  is  2,  i.e.  (  \/2)^  =  2. 
Again,  ^5  is  the  number  whose  cube  is  5,  i.e.  {^b)'^  =  5. 

240.  The  ?ith  root  of  a  number  which  is  not  a  perfect  nth 
power  is  called  an  incommensurable  root  or  an  irrational 
number;  as,  V2,  V3. 

241.  An  irrational  number,  or  any  other  number  which  is 
not  a  whole  or  a  fractional  number,  is  called  an  incommen- 
surable number ;  as  ^3,  -^6,  or  the  ratio  of  the  circum- 
ference of  a  circle  to  its  diameter. 

242.  Approximate  values  of  incommensurable  roots. 

If  to  2  we  add  ciphers  and  apply  the  method  of  finding 
the  square  root,  we  obtain  the  result  below : 

2.00000000)1.4142  ... 
1_ 
24)100  1st  remainder 

90 
281)400  2d  remainder 

281 
2824)11900        3d  remainder 

11290 
28282)60400    4th  remainder 
56564 
0.00003836    5th  remainder 


IRRATIONAL   NUMBERS  251 

Each  remainder  in  the  above  process  is  the  difference  be- 
tween 2  and  the  square  of  the  corresponding  part  of  the  root. 

This  remainder  decreases  rapidly  as  we  increase  the  num- 
ber of  figures  in  the  root;  hence  the  square  of  the  root 
found  approaches  nearer  and  continually  nearer  2;  and 
therefore  the  root  itself  approaches  nearer  and  continually 
nearer  ^2. 

By  continuing  the  operation  indefinitely  we  obtain  a  com- 
mensurable number  which  approaches  indefinitely  near  and 
continually  nearer  ^2,  but  which,  by  §  238,  can  never  reach 
^2.  This  increasing  commensurable  number  is  said  to 
approach  the  incommensurable  root  ^2  as  its  limit. 

In  like  manner  we  can  find  a  commensurable  number 
which  shall  differ  from  any  incommensurable  root  by  as 
little  as  we  please. 

Exercise  03. 

Obtain  to  three  places  of  decimals  the  value  of  the  roots : 

1.  V3-      4.   V^-       7.   V^-3-       10.   V^-^^-*-      13.   ^2. 

2.  V^-      5.   Vll-     8-    V^>-^-        11-   V^-^^-      14.   ^4. 

3.  V^.      6.   V13.     9.   V^-^^-      12-   V^-^-  15-   v^2.5. 

243.  The  quality-unit  + 1  or  —  1  multiplied  by  an  arith- 
metic incommensurable  number  is  a  positive  or  a  negative 
incommensurable  number  ;  as,   +  ^2,  —  ^3,  —  ^5. 

244.  The  fundamental  laws  which  have  been  proved  for 
commensurable  numbers  hold  also  for  incommensurable 
numbers.  The  proof  of  these  laws  for  incommensurable 
numbers  will  be  found  in  the  chapter  on  the  theory  of 
limits. 

245.  An  irrational  expression  is  one  which  involves  the  nth 
root  of  an  expression  which  is  not  a  perfect  nth  power ;  as, 
^^i  V(^  +  ^)-  ^^y  i^'^'^lional  numeral  expression  denotes 
an   irrational   number.      But,   just  as   a  fractional   literal 


252  ELEMENTS   OF  ALGEBRA 

expression  denotes  both  integral  and  fractional  numbers, 
so  an  irrational  literal  expression  denotes  both  rational  and 
irrational  numbers. 

E.g.,  the  irrational  literal  expression  ^a  denotes  a  commensurable 
or  rational  number,  when  a  =  1,  4,  9,  1/4,  4/9,  •••  and  an  incommen- 
surable or  irrational  number  when  a  =  2,  3,  5,  •••. 

Observe  that  commensurable  and  incommensurable  apply 
to  numbers  only  ;  while  rational  and  irrational  apply  to 
either  numbers  or  expressions. 

Exercise  94. 

1.  Is  the  number  -^jA  commensurable  or  incommensura- 
ble? sjQ?  V9?  V12?  V14?  V(4/9)?  ^(8/27)? 
-3/3?     -^216? 

2.  Is  the  expression  -^a*  rational  or  irrational  ?     -^x  ? 

^a^7     V(^  +  ^)^     ^{a  +  xy?     ^(a/x)? 

3.  Give  ten  sets  of  values  of  a  and  x  for  which  ^(a/x) 
denotes  a  commensurable,  or  rational,  number. 


CHAPTER  XVII 
SURDS 

246.  A  surd  number  is  an  irratioual  number  in  which  the 
radicand  is  a  rational  number;  in  other  words,  it  is  an 
incommensurable  root  of  a  commensurable  number. 

E.g.,  y/b,  ^7,  ^(2/3)  are  surd  numbers;  so  also  is  ^a  when  a 
denotes  a  commensurable  number  which  is  not  a  perfect  square. 

The  incommensurable  root  V(^  +  \/2)  i^  "^^  ^  surd  number,  since 
the  radicand  3  +  ^^2  is  not  a  commensurable  number. 

247.  A  surd  expression  is  an  irrational  expression  in 
which  each  radicand  is  a  rational  expression ;  as  -yja^  VS/G, 

248.  Surds  of  different  orders.  A  surd  of  the  second  order, 
or  a  quadratic  surd,  is  a  surd  with  the  index  2 ;  as  ^5,  ^a. 

A  surd  of  the  nth  order  is  a  surd  with  the  index  n  ;  as  -y/a. 
Observe  that  -^  ^5  is  a  surd  of  the  6th  order. 

249.  A  rational  number  or  expression  can  be  written  in 
the  form  of  a  surd  of  any  order. 

E.g.,  3  =  v^,  ^7,  {/81,  or  ^43 

and  «=  V«^  \/«^  v^«S  ^^  v^«^- 

250.  A  surd  expression  is  in  its  simplest  form  when  each 
radicand  is  integral,  its  numeral  factor  being  as  small  as 
possible,  and  its  literal  factor  of  as  low  degree  as  possible. 

E.g.,  the  simplest  form  of  the  surd  y/9>  is  2y/2. 
The  simplest  form  of  the  surd  ^(16  x*y5)  is  2  xy^(2  x^). 

263 


254  ELEMENTS   OF  ALGEBRA 

251.  Reduction  of  surds  to  their  simplest  form.  The  cases 
which  most  frequently  occur  are  the  three  following : 

I.  Radkand  integral.  Eesolve  the  radicand  into  two 
factors,  one  of  which  is  a  perfect  power  of  a  degree  equal  to 
the  order  of  the  surd,  and  apply  the  law,  ^{ab)  =  y/a  •  ^b. 

Ex.  1.  \/l35  =  v'PTs  =  3^5. 


Ex.  2.      7  V50  x'y<'  =  7  \/(5  x'^y)^  •  2  y 

=  7x5  x'^yV2y  =  35  x^yV2y, 


Ex.  3.    5  </l28xV  =  5  V  (4  xy)'^  -  2  x^y 

=  5  X  4xy  \/2  x'^y  =  20  xy  y/2  x^y. 

II.  Radicand  fractional.  Multiply  both  the  numerator 
and  denominator  by  such  a  number  as  will  make  the  de- 
nominator a  perfect  power  of  a  degree  equal  to  the  order 
of  the  surd,  and  apply  the  law,  ^{a/h)  =^a/^h. 

35      V35 


Ex.1.  :\^=;f^=^ 

\49      \  73         7 


sHba^x      </lbcfibx 


Ex   2      ^/  3a;   _  ^n^a%x  ^  V7^ 
\5a&5- \(5a6-0^         5 


ab^ 


III.     Factor  common  to  exponent  of  radicand  and  index 
of  root. 

In  this  case  we  apply  the  following  principle : 

The  index  of  the  root  and  the  exponent  of  the  radicand  can 
he  multiplied  or  divided  by  the  same  number. 

That  is,  "{/'a'"''  =  W«""'  =  V^""-  §§  223,  227 


Ex.  1.     ^(27  a^x^)  =  V(3  ax^  =  V(3  ax). 


Ex.  2.    ^(IG  a%i2)  =  ^x» .  (2  axy 


^x8.  </(i2axy  =  x^(2ax). 


SURDS  255 


Exercise  95. 

Reduce  each  oi 

nhet 

15. 

Allowing  surds  1 

:o  its 

%5. 

simplest  form ; 

1.    ^U7. 

■\/a^+2a^b+ab 

a    jb'-+^ 
6"\   a' 

2.    V288. 

16. 

Wh 

3.    3V150. 

17. 

ivf 

26. 

(«^^/:;:- 

4.  2V720. 

5.  ^256. 

18. 
19. 

3^|. 

27. 

6.  ^432. 

7.  5V245. 

20. 

3xy    j5z' 
z    \9a^y 

28. 
29. 

</25. 
</(8/:«0- 

8.  ^1029. 

9.  ^3125. 
10.    v'-2187. 

21. 
22. 

Sx  3/27 a* 
2a\   oj*   • 

26  J  a* 
a  \Sb' 

30. 
31. 

32. 

^(9/36). 

11.    V27a-^6^ 

12.    A/-108a.V- 

23. 

"aJ^3- 

33. 

13.    VarV""'. 

24. 

34. 
35. 

JJ/(32/x'»). 

14.  V.t-^-'Y^ 

•V(2V«"')- 

36.    '%/(S'"/x 

-). 

38. 

^c- 

-54x'"+'y'). 

37.    ^/{b-c 

W- 

c'*^).                  39. 

IJ/(64aV"). 

252.  Surds   which   are   rational   multiples  of   the   same 
monomial  surd  are  said  to  be  like,  or  similar,  surds. 

E.g.,  2y/3  and  y/'S/7  are  like  surds,  so  also  are  by/a  and  vTa,  or 
2y/a. 

253,  To  adci  o?*  subtract  surds,  we  reduce  them  to  their 
simplest  form,  and  unite  those  that  are  similar. 

Ex.1.    ^135  +  ^40  =  3^5  +  2^5  =  5^5. 
Ex.  2.   4\/T28  +  4V76  -  5^/1(32  =  32^/2  +  20^/3  -  45 V2 

=  20v3-  13  V2. 


256  ELEMENTS   OF  ALGEBRA 

Exercise  96, 
Simplify  each  of  the  following  surd  expressions : 

1.  V27+V48.  11.  V252-V294-48Vi. 

2.  2V180-V405.  12.  4V63  +  5V7-8V28. 

3.  2V28-V63.  13.  V^'  +  i^a'-3i^2U? 

4.  5V208-3V325.  14.  ^54+Vi-fVf 

3/orT     A.         3/^7-7   .      3 


5.  V^12-V50-V98.  15.    V27c^-V8c4+V125c. 

6.  3V12-V27  4-2VT5.       16.    "v/o^  - ^^«  + -V^326. 

7.  V^4-^V176  +  2V99.    17.    Va^+Vb^-^/Umc. 

8.  2V363-5V243  +  V192.    18.   3  V147  -  J  Vi  -  V2V 

9.  2^189+3^875-7^56.    19.   3Vl  +  3V|-A/|f 

10.    ^81-7^192+4-^648.    20.   j\-^72 -^^^ +  6^211. 

21 .  ^(9  aPf)  +  V(27  0^2/)  +  ^  a/(729  xV). 

22.  2  ^(3  a^fe)  -  -^(9  a%^  +  ^(125  a^d). 

23.  V(4<^'+4a'6)+V(9«&'  +  9&'). 


24.    Vit*^  —  a^2/  —  \^f—^  —  V (a;  +  2/) (^  —  2/0- 

254,  Surds  of  different  orders  can  be  reduced  to  identical 
surds  of  the  same  order.  This  order  can  be  any  common 
multiple  of  each  of  the  given  orders,  but  it  is  usually 
most  convenient  to  choose  the  least  common  multiple. 

Ex.  1.   Reduce  v^o^,  -v^P,  Vc^  to  identical  surds  of  the  same  order. 
The  L.  C.  M.  of  the  indices  3,  4,  and  6  is  12.     By  III.  of  §  251, 

V^^='<f^^;  </b^^'V¥;   V?='</g^o, 


SURDS  257 

Ex.  2.    Which  is  the  greater  ^6  or  ^10  ? 
Reducing  these  surds  to  the  same  order,  we  have 

^6  =  ?2/6*    =  1^1296,  (1) 

and  ^10  =  12/103  =  I^IOOO.  (2) 

From  their  values  in  (1)  and  (2),  it  follows  that  ^6  >  ^10. 

255.  The  product  of  two  or  more  surds  is  found  by  ap- 
plying the  law 

■;ya^b  =  ^{ab).  §  222 

Ex.- 1.    V7  X  v'28  =  V7  X  2  V7  =  2  X  7  =  14. 

Ex.  2.    2^14  XV21  =2V(14  x  21)=  2V(72  x  6)=  14^6. 

When  the  surds  are  of  different  orders,  they  should  be 
reduced  to  the  same  order. 

Ex.  3.    V3  X  ^2  =  ^38  X  ^2  =  ^(38  x  22)  =  ^108. 

Conversely  to  §  251,  the  coefficient  of  a  surd  can  be 
brought  under  the  radical  sign  by  reducing  it  to  the  form 
of  a  surd  of  the  same  order. 

Ex.  4.   5  V3  =  V25  X  V3  =  >/75. 

Ex.  6.   X  ^x^  =  {/7^  X  ^a;8  =  ^v?. 

Ex.  6.    Multiply  2^3  +  3^2  by  4v3  -  5^2. 
The  work  can  be  arranged  as  below  : 

2\/3  +  3V^ 

4\/3-5V2 


24  +  2V6-30  =  2V6-6. 


256.   In  finding  powers  of  monomial  surds  we  often  make 
use  of  the  law,  {-^aj  =  </ar  (§  226). 

Ex.1.    (3\/^)2  =  32(v/^)2  §119 

=  9^(ax)2  =  9^(a2x2). 

Ex.  2.    (2y/xy  =  2\y/xy  =  Sx^x. 


258  ELEMENTS   OF  ALGEBRA 

When  applicable,  the  identities  in  Chap.  IX.  should  be 
used  in  finding  the  products  of  polynomial  surds. 

Ex.  3.    ( V3  -  \/5)2  =  (  V3)2  -  2 V3  .  Vs  +  (V5)2 
=  3  -  2\/l5  +  5  =  8-  2V'l5. 

257.    Two  binomial  quadratic  surds  which  differ  only  in 
the  quality  of  a  surd  term  are  called  conjugate  surds. 

E.g.,  3  +  V^  3,nd  3  —  y/2  are  conjugate  surds  ;  so  also  are  y/a-\-  y/b 
and  ^a  —  y/b,  or  ^/a  +  y/b  and  -  ^/a  +  y/b. 

The  product  of  two  conjugate  surds  is  rational. 

E.g.,  (3  4-  V2)  (3  -  V2)  =  32  -  (  ^2)2  =  7. 

( V«  +  \/^)( v«  -  \/ft)  =  ( V«)^  -( V&)^  =  «  -  ^' 

Exercise  97. 
Eeduce  to  surds  of  the  same  order : 

1.  V3,  ^'^'  6-  V»?  ^^»  a/«- 

2.  V(V2).  ^(2/3).  7.  2,  ^3,  V4. 

3.  -^2,  -</3.  8.  ^3,  2,  ^7. 

4.  -^8,  V3,  -^6.  9.  V«^  ^^  Vc- 

Bring  the  coefficient  under  the  radical  sign  : 

10.    11 V2.  11.    14 V5.  12.    6^4.  13.    5^6. 

n^      4      177  _     3a5    /20  c^  _     2a   3'27  a^^ 

Which  is  the  greater : 

17.  V^  or  -3/10  ?  19.    5  V2  or  ^344  ? 

18.  V^oi^-v/H^  20.    ^5  or  ^10? 
21.    -^a'^  or  ya,  when  a<l?  when  a>l? 


SURDS  259 

Obtain   the    simplest   form    for   each    of    the    following 
products  : 

22.  2  V15  X  3  V5.  29.  ^168  x  ^147. 

23.  8V12  X  3V24.  30.  5^128  x  2^'432. 

24.  Vi2  X  V^T  X  V75-  31.  ViO  X  ^200. 

25.  ^16  X  ^6  X  ^9.  32.  ^4  x  V^'- 

26.  ^12 +-^75x^30.  33.  (V^-V3)(V^  +  V3)- 

27.  ^0x^12x^18.  34.  (V^)-V7)(V^'  +  V)"- 

28.  v^a;  +  2  X  -^x  -  2.  35.  ( V^  -  V-*^)  ( V^' +  V^)- 

36.  (  -  v'c  -  V«)  ( -  V^  +  V«)- 

37.  (-V^+V^)(V^+V»')- 

Find  each  of  the  following  powers : 

38.    (y/2f.  42.    (^aby.  46.  (v'^)«. 

39..  (2V3)*.  43.    (^ay.  47.  (2V^^)*. 

40.  (Va;)'.  44.    (-^'ay.  48.  (3^/^2^)*. 

41.  (-^ay.  45.    (-^63)^^  49.  (2^/^::rP)6^ 

50.  (V3-V5)^  54.    (^2-</4)2. 

51.  (4-2V3/.  55.    (V6-^2)^ 

52.  (vr^  4- 2  V3)=^.  56.    (V2+V3+V^)'- 

53.  (V3-V2)^  57.    (1+^2 +V^)'- 
Find  each  of  the  following  products,  and  simplify : 

58.  (2V5  +  3V3)(3V5-4V3). 

59.  ( V2  +  V^  +  V^)  (2  V2  +  3  V3  +  V^)- 

60.  (5+-^4)(V3+V2). 


260  ELEMENTS   OF  ALGEBBA 

61.  (2V3+^2)(2V3-^4). 

62.  (8-3V7)(8  +  3V7). 

63.  (H-V2-V3)'- 

64.  ( V2  +  V3  -  V^)  ( V2  +  V3  +  V5). 

258.   Division  of  surds. 

Suppose  it  is  required  to  compute  the  value  of  y/6/^7.  We  might 
find  ^6,  which  is  2.236.-.;  then  find  ^7,  wliich  is  2.645...;  and 
finally  divide  2.236  ...  by  2.645  .... 

Of  these  three  long  operations  two  will  be  avoided  if  we  first  multi- 
ply both  dividend  and  divisor  by  ^7,  as  below  : 

V5/V7=V3o/7 

=  5.916  •../7  =  0.845.... 

Observe  that  the  new  divisor  is  a  rational  number. 
This  example  illustrates  the  following  principle  : 

The  quotient  of  one  surd  divided  by  another  is  put  in  the 
simplest  form  for  computation  by  multiplying  both  divi- 
dend and  divisor  by  such  a  factor  as  will  render  the  divisor 
rational. 

This  process  is  called  rationalizing  the  divisor^  or  rational- 
izing the  denominator. 

The  factor  by  which  we  multiply  the  divisor  to  obtain  a 
rational  divisor  is  called  the  rationalizing  factor;  as,  -^7 
above. 

The  cases  which  most  frequently  occur  are  the  three 
following : 

I.   When  the  divisor  is  a  monomial  surd ;  as,  ^a;™. 

Ex.  1.     _J_^    2XV5    ^A 
3V5     3V5  X  V5     15 

Ex.  2.    _^  =  -A-^<J^=A>3. 
Here  the  rationalizing  factor  is  ^a*. 


8UBDS  261 

The  simplest   rationalizing  factor  of  -s/x"^  is  evidently 

II.  When  the  divisor  is  a  binomial  quadratic  surd,  the 
simplest  rationalizing  factor  is  the  conjugate  of  the  divisor. 

Ex    1    5+V7^(5+V7)(3+v7)^22  +  8v7^.  ^ 

•   3-V7      (3-V7)(3  +  V7)  9-7  ^' 

Ex.  2.    a/Q^  +  a/^ - C a/«  +  a/^)(V^  +  a/^) -- c<  +  2 Vo^  +  6 
V<^-V^     (V«  —  \/^)(v'«  +  V^)  a-b 

III.  When  the  divisor  is  of  the  form  (^a  -f-  -y/b)  +  -y/c, 
first  multiply  by  the  expression  (  y/'a  +  ^b)  —  ^c. 

The  divisor  thus  becomes 

( V«  +  V^)'  -  ( V^)'.  or  (a  +  6  -  c)  +  2  ^(ab).       (1) 
Next  we  multiply  by  the  conjugate  surd 

(a  4.  6  _  c)  -  2  ^(ab). 
The  divisor  thus  becomes  the  rational  expression 

(a-^-b-cf-  (2  Vaby. 

Ex.  ^^2 ^ V2(V2-fV3  4-V5) 

V2+V3-V5      [(V2+V3)-V'5][CV2+V3)  +  V5] 
^    2+V6  +  V/10    ^2  +  a/6+V10 

^  (2  +  yC  +  ylO)  X  V6 

_  2^/6  +  6  +  2^15  ^  v/6  +  3  +  v/15 
12  6 

259.   When  applicable,  the  identities  in  Chapter  IX.  should 
be  used  in  writing  the  quotient  of  two  binomial  surds. 

Ex.    V^^  +  Vy«_Cv/x)8  +  (Vy)'» 
=x-y/xy  -\-y. 


262 


ELEMENTS   OF  ALGEBRA 


Exercise  98. 
Compute  to  three  places  of  decimals : 

1.  14--V2.  3.   48--V6-  5.    144 --V6. 

2.  25 -V^-  4.    V2-V^-  6.    4--V243. 
Rationalize  the  denominator  and  simplify  : 

7.  3V3/(2V2).        11.    12/ V"^-  15-    V«/^«- 

8.  Vl5/V(3/5).       12.    2^6/V2.  16.    ^a/^a. 


9.    V21/V(V3). 
10.    10/^5. 


19. 


2V5 


V5  + V3 


20.    l^+li^^ 
15  -  2  V3 

21     V5  +  3V3. 

'    2V5-V3 


22. 


23. 


24. 


25. 


26. 


V6-3V12 

2V<3  + V12 

2  V3  +  3  V2 
5  +  2V6 


V9  + 


V9  4-  x'  +  3 


r 


£c  -f-  Vi^^  —  y'^ 

1 

1  +  V2  +  V^* 


13.    3V2/^9. 


17. 


V 


x'/^af. 


14.    -^20/(3^16).    18.    ^(ax)/^x. 
3 


27. 


2+V3  4-V5 


28.    ^+V^+V^. 
1+V2-V3 


29. 


30. 


31. 


32. 


33. 


34. 


V3 


V2+v^+V5 

3 

V2-V6-V'^* 


■Vx  —  2-\-Vx 


Va?  —  2  —  V  ic 


a— Va^  +  3 


Va  —  6  —  Va  +  b 


Va  —  &  4-  Va  +  b 

3  +  4V3 

V6+V2-V5' 


SUBDS  268 

Write  each  of  the  following  quotients : 

35.  (a  —  x)-i-  (y'a  —  ^x). 

36.  {ax-ay)/(^x-^^y). 

37.  (l-l/x)-(l4-l/V'^). 

38.  (a/b  -  x/y) h- [  V(« A)  +  V(V^)]- 

39.  (v«'-V^')^(V«-V^)- 

40.  (arV^J- W2/)-^(V^+V2/)• 
Rationalize  the  denominator  of  each  fraction : 


41. 


42. 


x" 


44 

Vl  +  ar^- 

-VI- 

X 

45. 

Vl  +  ar'  +  Vl- 
2  Va  +  6  —  3  Va 

a;- 

6 

Va^  H-  a*  H-  a  2  Va  +  6  —  Va  —  6 


43     V10+V5  4-V3.  46     (V3+ V5)(Vo  + V2) 

V3+V1<>-V5  *       V-+V3  +  V5 

260.   A  root  of  a  monomial  surd  is  found  by  applying  the 

^^^^  V  V«  =  V«-  §  227 

Ex.  1.    »/ V7  =  ^7. 

Observe  that  {/{fa=  {/{/a,  since  each  member  =  *^a. 

Ex.2.    Vv^(4x2)  =  {/V(4a;2)=  */(2x). 

Exercise  09. 
Simplify  each  of  the  following  expressions : 

1.  -^V(27a').      5.    V(^V^)-  9-    ^WV^)- 

2.  V^(9ajO-        6-    VA/(25iry/16).     10.    ^^--^(x/^x). 

3.  ^v(«0.        7.  V(^V^)-  11-  "-V(^/V^O- 

4.    ^^(A^)-        8.    V(2/V2)  12.    V(^V</^c)- 


264  ELEMENTS   OF  ALGEBRA 

PROPERTIES   OF   QUADRATIC   SURDS. 

261.  //  x+^y  =  a^^b,  (1) 

where  x  and  a  are  rational  numbers,  and  ^y  and  -^b  are  surd 
numbers;  then, 

X  =  a  and  y  —  b.  (2) 

Proof.    Transposing  a  and  squaring,  from  (1)  we  obtain 

(x  —  ay-\-2(x  —  a)-y^y-\-y  =  b. 

.'.  2(x  -  a)V2/  =  (b-y)-  (^  -  af.  (3) 

Since  a  surd  number  cannot  equal  a  rational  number, 
equation  (3)  is  satisfied  when  and  only  when  x  =  a  and 

y  =  ^• 

262.  A  quadratic  surd  number  cannot  be  equal  to  the  sum 
of  a  rational  number,  other  than  zero,  and  another  quadratic 
surd  number. 

Proof     Let  ^b  =  x-^^y,  (1) 

where  cc  is  a  rational  number  and  -^^b  and  -y/y  are  surd 
numbers;  then,  by  §  261,  we  have 

x  =  0  Siiid  y  =  b.  (2) 

263.  Square  root  of  the  binomial  surd  a  -\--^b. 

Suppose  Va  ±  V^  =  V^  ±  V2/-  (1) 

Square,  a  ±  Vb  =  x  +  y  ±  2Vxy. 

Hence,  by  §  261,  we  have 

x  +  y  =  a,  2-\/xy  =  -\/b ; 
or  x-\-y  =  a,      Axy  =  b.  (c) 

Solving  system  (c)  for  x  and  y,  and  substituting  their 
values  in  (1),  we  obtain  the  value  of  y'(a  ±  V^)- 

We  shall  here  consider  only  those  cases  in  which  system 
(c)  can  be  solved  by  inspection. 


SURDS  265 

Ex.  1.     Find  the  square  root  of  18  +  8^5. 


Assume  V18  +  8V5  =  V^  +  V'^-  (1) 

Square  IS -^S^o  =  x  +  y  +  2y/xy. 

.-.  a;  +  2/  =  18,  and  2>/^  =  8V5;  §261 

or  X  +  y  =  18,  and  xy  =  80.  (a) 

By  inspection  we  see  that  one  solution  of  system  (a)  is 
X  =  8,  y  =  10. 


.'.  V18  +  8^6  =  V8  +  VIO  =  2 V2  +  VIO. 
Ex.  2.     Extract  the  square  root  of  83  —  r2y'35. 


Assume       V83  -  12  ^35  =^x-  y/y. 

Square  83  -  12^35  =  x  +  y  -  2Vxy. 

.-.  X  +  y  =  83,  and  2Vxy  =  12 V35, 

or  x  +  y  =  83,  and  xy  =  1260. 

By  inspection,  x  =  63,  and  y  =  20. 

.-.  V83  -  12V35  =  V63  -  V20  =  3  V7  -  2y/5. 

By  taking  x  =  20  and  y  =  03,  we  would  obtoin  the  negative  root 
of  the  given  number. 

Exercise  lOO. 

Find  the  square  root  of  the  binomial  surds : 

1.  6  +  V20.              6.    11-2V30.  11.  4J-IV3. 

2.  12-6V3.            7.    7-2V10.  12.  17  -  2 V66. 

3.  16  +  6V7.            8.    17-12v'2.  13.  19+8V3. 

4.  13-2V42.          9.    47-4V33.  14.  11+4V6. 

5.  28-5V12.        10.    19  +  4V22.  15.  15-4V14. 


CHAPTER  XVIII 
IMAGINARY  AND   COMPLEX  NUMBERS 

264.  Quality-units  V—  1  and  —  V— 1.  A.s  we  have  seen 
in  §  219,  an  even  root  of  a  negative  number,  as  V—2, 
cannot  be  a  positive  or  a  negative  number,  and  therefore 
is  not  as  yet  included  in  our  number  system. 

To  give  a  meaning  to  such  expressions  as  V—  1  and 
V—  2,  we  assume  the  identity 

{</uy  =  u  (1) 

to  hold  when  u  is  negative  and  n  is  even  (§§  218,  239). 
Thus,  V—  2  denotes  that  number  whose  square  is  —2. 
An  important  particular  case  of  (1)  is 

(V~l)^  =  -l.  (2) 

Since  any  power  or  root  of  -f  1  or  —  1,  heretofore  ob- 
tained, is  a  quality-unit,  Ave  call  V—  1  a  quality-unit. 
That  is,  V  —  1  is  a  quality-unit  ivhose  square  is  ~  1. 
Squaring  both  members  of  (2),  we  obtain 

(V^iy^  +  l;  (3) 

that  is,  the  fourth  j^oiver  o/  V—  1  is  equal  to  -f  1. 

Again,  the  opposite  of  the  quality-unit  V—  1  is  —V—  1. 

Also,       (V"-^f-(V^'-V^l  =  -V^=^;  (4) 

that  is,  the  cube  o/  V—  1  is  equal  to  its  opposite,  —V—  1. 

The  quality-units  V—  1  and  —V—  1  involve  the  idea  of 
the  arithmetic  one  and  that  of  oppositeness  to  each  other. 

266 


IMAGINARY  NUMBERS  267 

Observe  carefully  the  above  relations  of  the  quality -unit 
V^  to  —  1,  +  1,  and  —  V^^.      

The  quality-units  V—  1  and  —  V  —  1  are  called  imaginary 
units. 

The  units  V—  1  and  —  V— 1  are  for  brevity  often  de- 
noted by  i  and  —  i,  i  being  used  as  a  numeral. 

Note.  The  word  imaginary,  as  here  used,  must  not  be  understood 
as  implying  that  the  units  i  and  —  i  are  any  less  real  than  -f  1  and 

—  1.  'J'he  expression  V—  1  was  called  imaginary  when  it  first  made 
its  appearance  in  Algebra  before  its  meaning  and  uses  were  under- 
stood. The  name  is  unfortunate,  but  with  the  above  explanation  we 
shall  use  it. 

265.  Imaginary  numbers.  Any  arithmetic  multiple  of  the 
imaginary  unit  i  or  —  i  is  called  an  imaginary  number. 

An  imaginary  number  is  commensurable  or  incommensura- 
ble, according  as  its  arithmetic  factor  is  commensurable  or 
incommensurable. 

E.g.,    t3,    t(3/5),     —  i7    are    commensurable,    while    i^y2    and 

—  1^(2/3)  are  incommensurable  imaginary  numbers. 

266.  Multiplication  by  the  imaginary  unit  V~  1'  ^^  ^  ^^ 
defined  by  assuming  the  commutative  law;  that  is,  a  being 
any  number,  we  assume  that 

a  X  V—  1  =  V—  1  X  a,  or  a  x  i  =  ia. 


267.    Since  /•«  =  «•  /,  the  imaginary  number  t  •  ci  or  —  /  •  a 
can  be  written  ai  or  —  ai. 

Imaginary  whole  numbers  form  the  following  series : 

...,     -3i,     -2i,     -i,     0,     i,     2i,     3*,     •••. 

Observe  that  the  one  and  only  number  which  is  common 
to  the  series  of  real  and  imaginary  numbers  is  0. 


268 


ELEMENTS   OF  ALGEBRA 


268.   Geometric  representation  of  quality-units. 

A  directed  line  is  a  line  whose  direction  and  length  are  botli  consid- 
ered.    To  represent  geometrically  quality-numbers  we  use  directed 

lines.     Of  the  directed  line  OA', 
■g  O  is  the  origin  and  A  the  end. 

Let  0^=  +  l;  then  0^'=-l. 

Substituting  these  values  in 

(+1).  V^ri.^/3-i  =  _i, 

we  obtain 

OA .  \/^  .  v"^  =  OA'. 

That  is,  multiplying  OA  by 
V—  1  twice  in  succession  re- 
verses its  direction ;  hence  we 
can  assume  that  multiplying  OA 
by  V—  i  twice  in  succession  re- 
volves   OA    through    two    right 

angles  in  the  plane  ABA'  and  in  a  direction  opposite  to  that  of  the 

hands  of  a  clock. 

Hence  multiplying  OA  by  yf^l  once  would  revolve  OA  through 

one  right  angle  in  this  direction  ;  that  is, 


/                        + 

/ 
/ 

f             ~1 

N 

"                              \ 
S 
\ 
\ 

\ 

\ 

+  '           1 

\       ^ 

\ 

N 

O    >       1 

/ 
/ 
/ 
/ 
/ 

y 

B' 


OA  '  V^l  =  OB. 

But  OA-V^l=(+l).V^^=V'^. 

From  (1),  (2),  OB  =  V^^,  or  i. 

.'.  OB'  =-  OB  =  -i. 


(1) 
(2) 


Hence,  if  the  primary  quality-unit  +  1  is  represented  by  the 
directed  line  OA,  the  quality-units  —  1,  i,  and  —  i  will  be  represented 
by  the  directed  lines  OA',  OB,  and  OB',  respectively. 

As  the  lines  OB  and  OB'  are  just  as  real  as  the  lines  OA  and  OA', 
so  the  quality-units  i  and  —  i  are  just  as  real  as  -|-  1  and  —  1. 

Arithmetic  multiples  of  i  and  —  i  can  be  represented  by  distances 
along  the  lines  OB  and  OB  or  their  extensions,  just  as  multiples  of 
-f  1  and  —  1  are  represented  by  distances  along  the  lines  OA  and 
OA'  or  their  extensions. 

Again,  if  in  a  football  game  we  denote  the  forces  exerted  in  the 
direction  OA  by  positive  real  numbers ;  then  negative  real  numbers 
will  denote  the  forces  exerted  in  the  opposite  direction  OA',  positive 


IMAGINARY  NUMBERS  269 

imaginary  numbers  will  denote  the  forces  exerted  in  the  direction  OB, 
and  negative  imaginary  numbers  will  denote  the  forces  exerted  in  the 
direction  OB'. 

To  express  by  numbers  the  magnitudes  and  directions  of  the  many 
other  forces  in  the  game  we  need  still  further  to  enlarge  our  concept 
of  quality-numbers,  as  is  done  in  §  285. 

269.  Since  imaginary  numbers  are  simply  arithmetic  mul- 
tiples of  the  units  i  and  —  /,  they  are  added  and  subtracted 
the  same  as  real  numbers. 

That  is,  at  ±  bi  =  {a±  b)i,  (1) 

which  is  the  converse  of  the  distributive  law. 

Ex.  1.    4  I  +  C  i  =  (4  4-  6)  t  =  10  I. 

Ex.  2.    (7/3)  i  -  (0/3)  i  =  (7/3  -  5/3)  i  =  (2/3)  i. 

270.  When  the  imaginary  unit  is  a  factor  of  a  product,  the 
distributive  law  follows  from  its  converse  in  §  269,  and  the 
associative  law  follows  from  the  commutative  law  in  §  266 ; 

that  is,  (fl  ±b)i=  ai  ±  bi,  §  269 

and  ai  •  bi  =  i-ab  =  -ab.  §  267 

271.  Powers  of  i.     From  §  264,  we  have 

,^  =  -1,  i'  =  ~i,  i*  =  -\-l.  (1) 

Ex.  1.     2-7    =  1-4  .  1-3  =  (+  1)  (  _  i)  =  _  f.  l)y  (1) 

Ex.  2.    I'lo  =(i4)--2i2  =(+  1)2(-  1)  =  -  1.  by  (2) 

Ex.  3.    zi8  =(^iiyi=(^\)Si:^  i^  by  ( 1 ) 

If  n  is  any  positive  integer  including  zero,  we  have 

,.4n     =(i4y  =  ^_^iy^_^i.  (2) 

/4«+i_^-4u^-  =/.  by  (2) 

/4n+2  =  ,;4«^-2  =  _  1  ;  by  (1),   (2) 

/4n+3  =  ^•4.^-3  ^  _  /  by   (1)^   (2) 


270  ELEMENTS  OF  ALGEBRA 

Hence^  any  eve7i  power  of  i  is  +1  or  —  1,  and  any  odd 
power  of  i  is  i  or  —  /. 

272.  The  square  root  of  any  yiegative  number  is  an  imagi- 
nary number; 

that  is,  V^  =  V«  •  '•  (1) 

Proof    By  the  commutative  and  associative  laws  we  have 

( V«  •  ^/^l)'  =  {^a)\-y/^^f  =  ~a;  §  271 

hence,  y^V^^  =  V~a,  or  conversely  (1).      §  221 

E.g.,  V-  16  =  i  .  vie  =  4  i,  or  W^l. 

and  y/—  ai=i '  Va^  ^  a^,  or  aV—  1. 

273.  To  add  or  subtract  imaginary  numbers  given  in  the 
form  Va,  we  first  reduce  them  to  the  typeform,  -^a  •  i. 

Ex.1.    V^l^  -i-y/-Sl-V-3{J  =  7i  +  9i-ei 

=  (7  +  9_6)i  =  10i,  orlOV^l. 

Ex.  2.    V-9a2  4.  V-  4  &-^  -  V-  7  c'^  =  3  a  •  i  +  2  6  •  i  -  c  V7  •  i 

=  (3a  +  26  -Cv/7)i. 

274.  From  the  commutative  and  associative  laws  we  have 
the  following  principle : 

The  product  of  two  or  more  quality-numbers  is  equal  to  the 
product  of  their  quality-units  into  the  product  of  their  arith- 
metic values. 


Ex.  1.  V-  5(- Vll)=  i  •-!  •  V^'  \/ll  =-  ^'V^^i  or  -V-  55. 

Ex.2.  V^8.  \A^  =  iV3- V7  =-V21. 

Ex.  3.  V^  .  V^ .  \/^  =  i3y'2  .  y'3  .  V5=  -  iy/^^,  or  -  V^M 

Ex.  4.  V^  •  V"^  •  v^  •  v/^  =  /I  V^  V^^  V^  V'  =  v^«^- 


IMAGINARY  NUMBERS 


271 


Exercise  101. 
Simplify  each  of  the  following  expressions : 

1.  V^^+V^=^-V^=loo. 

2.  V^^-V^^-hV^^^16. 

3.  v^^^-v^^-v^^n^. 


4.    V-l)a^-V-4a2_V-l(5a='. 


5.    V -  36  6^  _  V -  49  6^  _f-  V -  81  6^ 


6.    V  —  (a;  +  af.+V—  {x  —  af. 


7.    3  V^  +  7  V-c  -  11 V-6  +  2  V-(m-?r)- 
8.    V2V^^.  12.    V^^-V^=^. 

9.  V3-V^^.  13.  V^^Vie. 

10.  V^^  •  V^ni.  14.    V^=l^  •  V^3  . 

11.  v^^V^.  15.  Ve-V^^-V 

16.  V5 .  V^=^  •  V^=^ .  V^=^. 


-5. 


17.  V^=^'  •  V^=^'  •  V~^^. 

18.  V—  3 ax  .  V—  3  6u;  •  V—  4  aft. 

19.  (V:r2  +  v^35)(V^^+V^. 

20.  (2  V"=^  +  3  \/^=^)  (5  V^^  -  2  V^=^). 

21.  (V— a;  +  V— y)(V— a?— V^^). 
24.    V^^       25.    (V^)^ 


22.    V-rt'-^.       23.    ( 


6)^ 


275.   Quotient  of  one  quality-unit  by  another. 

/^/=l-    /-^(+l)^/;    /^  (_!)=_/;  §84 

+  1  ^/=/^--/  =  r^  =  -/;  §  2G4 

—  1  -i-  /  =  i^  -^  ij  =  i. 


272  ELEMENTS   OF  ALGEBRA 

E.g.,      1  ^  i5  =  1  ^  i  =  _  i  ;   _  1  ^  i6  =  _  1  ^  (_  1)  =  1 ; 
i  ^  f  =  i  -^ (—  i)  =  —  I. 

276.  From  the  commutative  and  associative  laws  we  "have 
the  following  principle : 

The  quotient  of  two  quality-numbers  is  equal  to  the  quotient 
of  their  quality-units  into  the  quotient  of  their  arithmetic 
values. 

Ex  1      V^5  _i       V5  _  V2. 

Ex.2.   ^^.A.^  =  ,^,  or^^^^. 

V7        +1      v/7  7    '  7 

Ex.  3.    y/~a/  \/~b  =  (i/i)  ( y/a/y/h)  =  \/a/b. 

Ex.4.    y/^/V^  =  (+l/i)i^a/^b)  =  -iVa7b,  or  -\/-(a/6). 

Exercise  102. 
Perform  the  operation  of  division  in 
1.    i^  -r-  i.  S.    1  -r-  i^  5.    i  -J-  ^^ 


2.    ^5^^•.  4.    - 1 -m'^.  6.    V-14-V-2. 

7.    V^^ie^V^l.  11.    V^'-^V^'. 


8.  V^=n^^V^^.  12.   (V=T2-V-15)-r-V-3. 

9.  V^^^V~a.  13.  (V~a+V--6)  ^V~c. 
10.  V^(^)-^V£c.  14.  (Vl6-V8)-=-V^^. 
15.  i-v-i^l       16.    -i-~i''.       17.    ^3-^F.       18.    i*^  _j.  jT-ss^ 

COMPLEX  NUMBERS. 

277.  The  sum  of  a  real  number  and  an  imaginary  number 
is  called  a  complex  number,  as  4:  ±  5  i,  7  ±  3  i. 

The  general  expression  for  a  complex  number  is  evidently 
a  +  bi,  where  a  and  b  are  any  real  numbers. 


COMPLEX  NUMBERS  273 

When  6  =  0,  a-\-hi  =  a,  a  real  number. 

When  a  =  0,  a-\-hi  =  hi,  an  imaginary  number. 

278.  We  define  addition  of  complex  numbers  by  assum- 
ing the  commutative,  and  therefore  the  associative,  law  of 
addition  for  real  and  imaginary  numbers. 

Hence,  in  adding  or  subtracting  complex  numbers  the  real 
parts  can  be  added  or  subtracted  by  themselves  and  the 
imroginary  parts  by  themselves. 

That  is,  (a  +  bi)  ±  (c  +  di)  =  (a  ±  c) -\- (b  ±  d)  L  (1) 

Ex.  When  is  the  second  member  of  ( 1 )  a  complex  number  ?  When 
an  imaginary  number  ?     When  a  real  number  ? 

279.  If  two  complex  numbers  are  equal,  their  reed  parts  are 
equal,  and  their  imaginary  parts  are  equal. 

Proof.     Let  a-{-bi  =  c-{-di,  (1) 

where  a,  b,  c,  d  are  all  real  numbers. 

Transposing,  a  —  c  =  di  —  bi.  (2) 

But,  if  a  real  number  is  equal  to  an  imaginary  number, 
each  is  zero  (§  267) ;  hence, 

a  —  c  =  0,  or  a  =  c, 

and  di  —  bi  =  0,  or  d  =  b. 

An  important  case  of  this  theorem  is  the  following : 
If  a  +  bi  =  0,  then  a  =  0  and  6  =  0. 

280.  Two  complex  numbers  which  differ  only  in  the 
signs  before  their  imaginary  terms  are  called  conjugate  com- 
plex numbers,  as  a  -f  bi  and  a  —  bi. 

Since  (a  +  bi)  -\- {a  —  bi)  =  2  a, 

the  sum  of  two  conjugate  complex  numbers  is  real. 


274  ELEMENTS   OF  ALGEBRA 

281.  Multiplication  by  a  complex  number  is  defined  by 
assuming  the  distributive  law ;    tliat  is, 

(a  H-  bi)  (c  +  di)  =  ac  -\-  adi  -f  bci  +  bdv^ 

=  (ac  —  bd)  4-  (ad  -f-  be)  i.  (1) 

Before  multiplying  one  complex  number  by  another  it  is 
convenient  to  reduce  each  to  the  type-form  a  -f-  bi. 

Ex.     (3  +  V^^)(4  -  V^S)  =  (3  +  v^  •  0(4  -  V''^  •  0 

=  12 +(4^5- 3^3)1 -fVlS- 

282.  From  (1)  in  §  281,  it  follows  that  the  product  of  two 
complex  numbers  is,  in  general,  a  complex  number. 

But,  the  produet  of  two  eonjugate  eomplex  numbers  is  real 
and  jjositive. 

Proof.      (a  +  bi)  (a  -  bi)  =  a^  -  {bif  =  a^  +  b\ 

E.g.,  (-3  +  V^^)(-3-V^^)  =  (-3)2-(V32)2:=ll. 

283.  The  quotient  of  one  complex  number  by  another  is,  in 
general,  a  complex  number. 

Proof  a  +  bi_(a  +  bi)(c-di) 

c  -f  di"~~  (c  H-  di)  (c  —  di) 

_ac  +  bd      be  —  ad  .  ^-js 

~  c^  +  d^  -^  c2  +  d=^**  ^^ 

284.  From  (1)  in  §  283,  it  follows  that  when  the  divisor 
is  a  complex  number,  the  quotient  can  be  expressed  as  a 
complex  number  by  multiplying  both  the  dividend  and  divi- 
sor by  the  conjugate  of  the  divisor. 

Ex  4  +  3  I  ^  (4  +  3  0(3  +  20 

S-'2i      (3-2  0(3  +  20 

^6  +  17j^A  +  iIj- 
9  +  4        13      13  ' 


COMPLEX  NUMBERS 


275 


Exercise  103. 
Find  each  of  the  following  sums  and  products : 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 

19.  (V^^4-V2) 

20.  (2  +  3aO'- 


2+V-4)  +  (3-V-l). 


3+V-9)-(7-V-16). 
3-2  0 +  (6 +  0  1). 


8+V-36)-(o+V-25). 
4  +  3  0  -  (3  -  4 1). 

2+V^  +  (2-V^:r3). 

2-hV^^)(3-V^. 


4+V-16)(3-V-25). 
5  +  2  V^  (2  -  3  V^l). 


l+V-7)(2-V-T6). 
3-hv'^'5)(2-V^. 
V2  +  V^2)  ( V3  -  V^. 
2+V^(2-V^. 
_4-V^(-4+V^. 
_7-V^^ll)(-7+V^ll). 
.rV-  X  +  2/V^)  (xV^^  -  2/V^). 
V^T  4.  5V^3)(V^^  +  3V^2). 

21.  (2-3V^2)3. 

22.  (a  -f  ci)l 


276 


ELEMENTS   OF  ALGEBBA 


Reduce  each  of  the  following  expressions  to  the  typical 
form  a  -\-hi: 


23. 


24. 


25. 


1-V-l 


26. 


27. 


28. 


a  —V—  X 
1 

3  4-4V^5 
2_3V^^ 


29. 


30. 


31. 


3+2V-1 
2  -  3V^=~1 

3-4^' 


2-V-2 
Show  that 

3)/2]»  =  +  1 ;  also  [(-  1  -  V=3)/2]-'  =  +  1 ; 
and  that  therefore  there  are  at  least  three  cube  roots  of  +1. 

33.  In  §  281,  when  is  the  product  a  complex  number  ? 
When  an  imaginary  number  ?     When  a  real  number  ? 

34.  In  §  283,  when  is  the  quotient  a  complex  number  ? 
When  an  imaginary  number  ?     When  a  real  number  ? 

285.   Geometric  representation  of  a  complex  number. 

To  find  the  sum  of  +  4  and  —  1  geometrically,  we  lay  off  Oil/ equal 
to  4  in  the  positive  direction  ;  then  from  the  end  of  OM  we  lay  off  MB 

equal  to  1  in  the  nega- 
tive direction.  The 
straight  line  OB^  which 
extends  from  the  origin 
0  of  the  line  03/ to  the 
end  B  of  the  line  MB, 
is  the  sum  of  +  4  and 
—  1  ;  that  is,  the  sum 
is  the  directed  line 
dravm  from  the  origin 
of  the  first  line  to  the 
end  of  the  second. 

Similarly,    to    con- 
struct the  sum  4  +  3 1, 
we  lay  off  OM  equal  to 
+  4  ;  at  il/,  the  end  of  OM,  we  erect  a  perpendicular  and  on  it,  in  the 


COMPLEX  NUMBERS  277 

direction  of  positive  imaginaries,  lay  off  MP  three  units  long ;  then 
OP,  i.e.  the  directed  line  extending  from  the  origin  O  of  the  first  line 
to  the  end  P  of  the  second  line,  will  represent  the  complex  number 
4  +  3  i. 

From  the  right-angled  triangle  OMP  we  have 


the  length  of  OP  =  \AP  +  3^  =  5. 

If  we  take  011=  one  unit  long  and  draw  HR  parallel  to  MP,  then 
from  the  similar  triangles  Olill  and  OMP  we  have  OB  =  4/5  and 
RH=  (3/5)i,  that  is,  the  directed  unit-line  0^  represents  4/5+  (3/5)2, 
and  OP,  which  represents  4  +  3  j,  is  5  times  this  unit-line. 

Hence  the  arithmetic  value  of  4  +  3  i  is  5,  and  its  quality-unit  is 
4/5  +(3/5)/,  which  illustrates  §  286. 

Observe  that  the  quality-unit  4/5 +(3/5)i  is  obtained  by  dividing 
4  +  3  I  by  5,  and  that  5  =  \/42  +  32. 

In  like  manner  we  can  represent  any  other  complex  number  and  its 
quality-unit. 

286.  The  arithmetic  value,  or  modulus,  of  the  complex  num- 
ber a  +-  hi  is  the  square  root  of  the  sum  of  the  squares  of  a 
aud  6,  or  Va^  +  ^^;  and  its  quality-unit  is  (a  -f  hi)/-\/a--\-h^. 


That  is,  a-irU  =  [(a  +-  hi)/^a?  -f  6^]  .  Va'  +  h\ 

where  the  complex  nnmber  a  -f  hi  is  written  as  the  product 
of  a  quality-unit  and  an  arithme:tic  number. 


E.g.  the  arithmetic  value,  or  modulus,  of  4  +  3i  is  v42  +  3^,  or  5. 


The  modulus  of  5  -  3  i  is  V62+(-3)2,  or  y/ZA. 


The  modulus  of  -  1/2  +  \/^/2  is  V(l/2)2 +  (V3/2)2,  or  1. 


CHAPTER   XIX 

QUADRATIC  EQUATIONS  IN  ONE   UNKNOWN 

287.  By  the  principles  of  equivalence  of  equations  in 
Chapter  VII.  we  can  derive  from  any  quadratic  equation  in 
one  unknown,  as  x,  an  equivalent  equation  of  the  type-form 

ax'  +  bx  +  c  =  0.  (A) 

Observe  that  in  (A),  ax-  is  the  sum  of  all  the  terms  in  .^^, 
bx  is  the  sum  of  all  the  terms  in  x,  and  c  is  the  sum  of  all 
the  terms  free  from  x. 

E.g.,  from  the  quadratic  equation 

6  x^  -  3  _  Gx^-S  ^  x2-  lOx 
2  4  8 

we  derive  tlie  equivalent  equation 

13  x2  -  40  X  -  9  =  0,  (1) 

which  is  in  the  type-form. 

Comparing  (1)  with  (A)  we  liave 

a  =  13,  6  =  -  40,  c  =  -  9. 

If  a  =  0,  (A)  ceases  to  be  a  quadratic  equation ;  hence  in 
what  follows  we  shall  assume  that  a  is  not  zero. 

If  neither  b  nor  c  is  zero,  (^1)  is  called  a  complete  quad- 
ratic equation. 

If  either  b  or  c  is  zero,  or  if  both  are  zero,  {A)  is  called 
an  incomplete  quadratic  equation.  When  6  =  0,  the  incom- 
plete equation  is  often  called  a  pure  or  binomial  quadratic 
equation. 

E.g.,  equation  (1)  is  a  complete  quadratic  equation;  while 
3  x2  +  4  X  —  0,  8  ic2  +  9  =  0,  and  -5  x^  =  0  are  incomplete,  the  last  two 
being  pure. 

278 


QUADRATIC  EQUATIONS  279 

Ex.  1.  In  examples  20-30  of  exercise  105  reduce  each  equation  to 
an  equivalent  equation  of  the  type-form  (J.),  and  state  the  values 
of  a,  6,  and  c  in  each. 

Ex.  2.    Solve  the  incomplete  quadratic  equation 

ax:^-^bx  =  0.  (1) 

Factor  x  (ax  +  6)  =  0.  (2) 

Equation  (2)  is  equivalent  to  the  two  linear  equations, 

X  =  0,  ax  -\-  b  =  0. 
Hence  the  roots  of  (2)  or  (1)  are  0  and  —  b/a. 
Remember  that  to  solve  any  quadratic  or  higher  equation  we  must 
first  find  its  equivalent  linear  equations.     Reread  §§  148  and  149. 

288.  Since  w^  +  mu  +  {m/2f  =  (u  +  m/2)2,  §  137 

TJie  expression  u^  +  mu  is  made  a  perfect  square  by  adding 
{m/2y,  or  the  square  of  one-half  the  coefficient  of  u. 

The  addition  of  (m/2y  is  called  completing  the  square. 

E.g. J  x2-7x  is  made  a  perfect  square  by  adding  (—7/2)"^  or 
(7/2)-^ 
that  is,  x'^-lx-h  49/4  =  (x-  7/2)2. 

4a;2  +  8x,  or  (2x)^  +  4(2x),  is  made  a  perfect  square  by  adding 
(4/2)2,  or  4 ; 
that  is,  (2  x)2  +  4  (2  x)  +  4  =  (2  x  +  2)2. 

289.  Any  quadratic  equation  can  be  solved  by  transposing 
all  its  Jtenns  to  one  member,  factoring  that  member  by 
vmting  it  as  the  difference  of  two  squares,  and  then  putting 
each  factor  equal  to  zero. 

The.  following  examples  will  illustrate  the  method. 

Ex.  1.   Solve  the  pure  quadratic  equation 

«X2  +  C  =  0.  (1) 

Divide  by  a,  a:2  _(- c/a)  =  0.  (2) 


Factor,  (x-V- c/a)(a;  +  v -c/a)=  0. 

By  §  149,  oj  =  V— c/a,  X  =  — V— c/a. 


280  ELEMENTS  OF  ALGEBRA 

Writing  these  two  linear  equations  together,  we  have 

x  =  ±  V-  c/a.  (3) 

Ex.  2.    Solve  the  complete  quadratic  equation 

x^  +  4x-2  =  0.  (1) 

Add  4-4,  a;2  +  4a;  +  4-6  =  0; 

or  (x  +  2)2-(V6)2  =  0.  (2) 

Factor,  (a;  +  2  -  VC)  (x  +  2  +  V6)  =  0. 

By  §  149,  x  +  2  =  V6,  x  +  2=-V6. 

Writing  these  two  linear  equations  together,  we  have 

x  +  2  =  ±V6.  (3) 

,',  x  =  -2±  V6. 

Observe  that  in  each  example  the  two  linear  equations  in  (3)  can 
be  obtained  from  (2)  by  transposing  the  known  term  and  then 
extracting  the  square  root  of  both  members,  writing  the  double  sign  ± 
with  one  member.  The  principle  of  equivalence  of  equations  which 
this  illustrates  is  proved  in  the  next  article. 

290.  Square  root.  If  the  square  root  of  both  members  of  an 
equation  is  extracted,  and  the  double  sign  ±  is  written  before 
one  member,  the  two  derived  equations  (when  rational  in  the 
unknown)  will  together  be  equivalent  to  the  given  equation. 

Proof     Let  the  given  equation  be 

A'  =  B^,  (1) 

Avhere  A  and  B  are  rational  in  the  unknown. 

Transpose,  A^-B^  =  0. 

Factor,  (A  -  B)  {A  +  5)  =  0.  (2) 

By  §§  149  and  106,  (2)  is  equivalent  to  the  two  equations 

A  =  ±B.  (3) 


QUADRATIC  EQUATIONS  281 

Equations  (3)  can  be  obtained  from  (1)  by  extracting  the 
square  root  of  both  members  and  writing  the  double  sign  ± 
with  the  second  member.;  hence  the  theorem. 

The  following  examples  illustrate  how  this  principle, 
which  is  proved  by  factoring,  abbreviates  the  work  of 
finding  the  two  linear  equations,  which  are  equivalent  to 
a  given  quadratic  equation. 

Ex.  1.     Solve  x2  +  32  -  lOx  =  0.  (1) 

Transpose  32,  a;2  -  10  x  =  -  32. 

Add  (10/2)-2,  x2  -  10  «  +  25  =  25  -  32  =  -  7. 

Extract  square  root,  x  —  5  =  ±  V—  7. 

.-.  X  =  5  ±  V^.  (2) 

By  §§  106  and  290,  no  root  is  either  introduced  or  lost  in  passing 
from  (1)  to  (2);  hence  the  roots  of  (1)  are  5  +  V—  7  and  5—  V—  7- 

Ex.2.     Solve  xM:_9^xM:J^  ,j^ 

4  5 

Multiply  by  20,  5  x*  +  45  =  4  x^  +  4. 

Transpose,  x^  =  —  41. 

Extract  square  root,  x  =  ±  V—  41.  (2) 

By  §§  106,  108,  and  290,  no  root  is  either  introduced  or  lost  in 
passing  from  (1)  to  (2);  hence  the  roots  of  (1)  are  +  V—  41  and 

Ex.3.     Solve  2x2  =  7x  +  ll. 

Transpose  7  x,  2  x^  —  7  x  =  11. 

Multiply  by  2,  (2  x)2  -  7  (2  x)  =  22. 

Complete  square,   (2  x)2  -  7  (2  x)  +  49/4  =  22  +  49/4  =  137/4. 

Extract  square  root,  2  x  -  7/2  =  ±  ^1^7/2. 

:  .-.  x=(7±Vl37)/4. 

Hence,  to  solve  a  quadratic  equation  we  can  proceed  as 
follows : 
Reduce  the  equation  to  the  form  aa^  -\-bx  =  —  c. 


282  ELEMENTS    OF  ALGEBRA 

If  the  tei'm  in  xr  is  not  a  perfect  square,  multiply  (or 
divide)  both  members  by  a  number  which  will  make  it  a 
perfect  square. 

Add  to  both  members  what  is  necessary  to  complete  the 
square  of  the  unknown  member. 

Extract  the  square  root  of  each  member,  writing  the 
double  sign  ±  before  the  known  member. 

Solve  the  two  derived  linear  equations. 

Exercise  104. 
Solve  each  of  the  following  equations : 

1.  x'-Jrl  =  4.x.  9.  3x'-6x-\-2  =  0. 

2.  x^-2x  =  4:.  10.  5x--6x-\-ll  =  0. 

3.  ^'-^  +  5  =  80;.  11.  3^2 +  4 a;  + 7  =  0. 

4.  'x''-{-2x  =  2.  12.  2x'-6x-{-10  =  0. 

5.  x'^-\-6x  =  -S.  13.  5x'-{-Sx  +  21  =  0. 

6.  4j^  +  4ic  =  ll.  14.  2x'-5x-\-15  =  0. 

7.  9x^-\-6x  =  17.  15.  2x'-3ax^2a'  =  0. 

8.  Ax^-4.x-7  =  0.  16.  {x-7y  =  i9(x  +  2y. 

When  both  members  are  perfect  squares  in  the  unknown,  as  in 
example  16  (or  can  be  made  so,  as  in  some  of  the  examples  which 
follow),  the  first  step  is  to  extract  the  square  root  of  both  members. 

17.  {x-{-2y  =  4.(x-lf.  21.  x''-\-2ax  =  b^-}-2ab. 

18.  (x+6y  =  16(x-6y.  22.  x' -{-2  ab  =  b^ +  2ax. 

19.  (ic  +  8f  =  9i»l  23.  4.x^-\-4:ax  =  b'-a\ 

20.  x^-3ax-\-2a^  =  0.  24.  x'^  +  3  a^  =  4.  ax. 

291.    To  solve  the  general  quadratic  equation, 

ajr^  +  6jr  +  c  =  0,  {A) 

we  proceed  just  as  with  the  particular  equations  above. 
Transpose  c,  ax^  -\-bx  =  —  c. 


QUADRATIC  EQUATIONS  283 

Multiply  by  4  a  instead  of  a,  to  avoid  fractions  in  (1) 
and  (2), 

(2  axf  -\-2b(2ax)  =  -4:  ac. 

Add  b',     (2 axy -\-2b(2ax)-l-b'  =  b'-4.ac.  (1) 


Extract  square  root,       2ax-\-b  =  ±  V6-  —  4 ac.  (2) 

Hence,  x  =  (-b±  Vb'-^ac)  /(2  a).  (B) 

By  §§  106,  108,  and  290,  no  root  is  either  introduced  or 
lost  in  passing  from  (A)  to  (B) ;  hence  the  roots  of  (A)  are 
given  in  (B). 

Let  b'  and  c'  denote  the  values  of  b  and  c  when  a  =  l. 
/Ihen  when  a  =  1,  equations  (^1)  and  (B)  become 


and  x  =  -b '/2  ±  V{b'/2f  - c.  (B') 

By  §  287,  any  quadratic  equation  can  be  reduced  to  an 
equivalent  equation  of  the  form  (A)-,  hence,  a  quadratic 
equation  in  one  unknown  has  ttvo,  and  only  two,  roots. 

292.  Solution  by  formula.  Instead  of  repeating  the  pro- 
cess in  §  291  with  every  quadratic  equation,  we  should  here- 
after find  the  values  of  a,  b,  and  c  when  the  equation  is 
reduced  to  the  type-form  {A),  and  substitute  these  values  in 
the  two  equations  (5), 

x  =  {-b±  V6^-4ac)/(2  a).  {B) 

Ex.  1.    Solve  2  X'^  -  3  X  -I-  5  =  0. 
Here  a  =  2,  6  =  —  3,  c  =  5. 
Substituting  these  values  in  equations  (  B) ,  we  obtain 


X  =  (3  ±  V9  -  40) /4  =  (3  ±  V-  31)/4. 
Ex.  2.    Solve  -  3  x2  =  3  A:  -  2  ax. 
Here  a=  —  3,  6  =  2  a,  c  =  —  ^k. 
Substituting  these  values  in  equations  {B),  we  obtain 


X  =(-  2  a  ±  \/4  a^  -  3(5  k) / {-  6) 
=  (aTVa2^^^9lfc)/3. 


284  ELEMENTS   OF  ALGEBRA 

293.  Equations  (JB')  of  §  291  afford  the  following  simple 
rule  for  writing  out  the  two  roots  of  an  equation  in  the  form 

Tlie  two  roots  are  equal  to  minus  one-half  the  coefficient  of  x 
plus  and  minus  the  square  root  of  the  binomial,  the  square  of 
one-half  the  coefficient  of  x  minus  the  known  term. 

Ex.  3.   Solve  ic2  +  4  a;  +  7  =  0,  by  the  rule  given  above. 

x  =  -2±  \/22  -  7  =  -  2  ±  V^. 

Ex.  4.    Solve  x2  -  6  a;  -  8  =  0, 

a:  =  3  ±  V(-  3)2  -(-  8)  =  3  ±  V17. 

Exercise  105. 
Solve  each  of  the  following  equations  by  §  293 : 

1.  cc2-2a;  =  l.  7.    a^  +  31  =  10aj. 

2.  a^  +  8a;  +  5  =  0.  8.    «2  +  6a;  +  ll=0. 

3.  a;2_^4a;  =  l.  9.    aj^  +  10 a?  +  32  =  0. 

4.  a;2  +  18  =  10a;.  10.    x'-\-52  =  Ux. 

5.  x'-{-3  =  2x.  11.    x'-\-2x  =  l. 

6.  a^  +  ll  =  4a;.  12.    x^  =  4:X-lS. 

Solve  each  of  the  following  equations  by  §  292 : 

13.  3a^  +  121  =  44x.  19.  21  +  0^  =  2  a^^ 

14.  25x  =  6x^  +  21.  20.  9a^-U3  =  6x. 

15.  8af-\-x  =  S0.  21.  12 a;^  =  29 a; -  14. 

16.  3a;2  +  35  =  22a;.  22.  20a;2  =  12-a;. 

17.  x-{-22  =  6x\  23.  15  aj^  -  2  aa;  =  a^. 

18.  15  =  17 a;  +  4a;2.  24.  21  x^  =  2 ax -\- 3 a'. 


QUADRATIC  EQUATIONS  285 

Solve  each  of  the  following  equations  by  the  method  best 
suited  to  it : 

25.    9x'-eax  =  a~-b'.  26.    aix" -{-!)  ^x^a" +  1). 

27.  a(pc^-l)-\-x(a^-l)=0. 

28.  x'-2(a-b)x-]-h^  =  2ab. 

29.  (b  —  c)x^+(c  —  a)x  =  b  —  a. 

30.  (a  4-  6)  a^  4-  c.r  =  a  +  &  +  c. 

31.  abx^-(a^-{-b^x  +  ab  =  0. 

32.  (a^  -  62)  (a:2_  1)^4^52.^ 

33.  {b''-a'^(x'  +  l)  =  2{a'-hb^x, 

34.  (a-a;)3  +  (a;-6)»  =  («-6)l 

35.  (a;-a  +  2&)3-(a;-2a  +  6)'  =  (a4-&)^ 


294.  Discussion  of  the  roots,  (—  b  ±  VA^  —  4  ac)/(2  a), 
wlien  a,  6,  c  are  real. 

(i)  If  6^  —  4  ac  >  0,  the  two  roots  will  be  real  and  unequal. 

(ii)  If  6^  —  4  oc  =  0,  the  two  roots  will  be  real  and  egzmZ. 

(iii)  If  6^  —  4  ac  <  0,  the  two  roots  will  be  imaginary  or 
complex. 

(iv)  If  6  =  0,  the  two  roots  will  be  both  real  or  both  imagi- 
nary, but  opposite  in  quality  and  arithmetically  equal. 

(v)  If  c  =  0,  one  root  will  be  zero  and  the  other  —  b/a. 

(vi)  If  6  =  c  =  0,  both  roots  will  be  zero. 

(vii)  Both  roots  will  be  real,  both  imaginary,  or  both 
complex. 

(viii)  If  br  —  4  ftc  is  a  perfect  square,  the  two  roots  will 
be  rational  when  a  and  b  are  rational. 

The  pupil  should  give  the  reasons  for  each  of  the  above 
statements. 


286  ELEMENTS   OF  ALGEBRA 

Ex.  1.    What  kind  of  numbers  are  the  roots  of  the  equation 

Sx'^ -2x^-1?  (1) 

Here  a  =  3,  6  =  -  2,  c  =  7  ; 

...  62  _4(^c  =(-2)2-4.3.7<0. 
Hence  the  roots  of  (1)  are  complex  and  unequal. 

295.   Sum  and  product  of  roots  of  jr-  -h  6'jr  +  c'  =  0.         (A') 
Representing  the  two  roots  of  (A')  by  Xi  and  x^,  we  have 


x,^-b'/2  +  V(b'/2y-c,  (1) 


X,  =  -  b'/2  -  ■\/(b'/2y  -  c.  (2) 

Adding  (1)  and  (2)  to  find  the  sum,  we  obtain 

x,  +  x,  =  -b'.  (3) 

Multiplying  (1)  by  (2)  to  find  the  product,  we  obtain 

<c,.x,=  (-  672)^  -  [(672)^  -  c]  =  c.  (4) 

Hence,  if  a  quadratic  equation  is  in  the  form 

x'-\-b'x-\-c'  =  0,  (A') 

the  sum  of  its  roots  is  equal  to  minus  the  coefficient  of  x,  and 
the  product  of  its  roots  is  equal  to  the  known  term. 

E.g.,  the  equation  3  x^  =  7  x  +  5  put  in  the  form  of  {A')  becomes 

x^-^x-^  =  0. 
Hence  the  sum  of  the  roots  is  7/3,  and  the  product  is  —  5/3. 
Note  that  this  principle  agrees  with  §  139  in  factoring,  and  that  it 
is  in  reality  only  another  form  of  stating  the  principle  in  that  article. 

Exercise  106. 

1.  By  §  294,  what  kind  of  numbers  are  the  roots  of  each 
of  the  equations  from  5  to  14  in  exercise  104  ? 

2.  By  §  295,  what  is  the  sum  and  what  the  product  of 
the  roots  of  each  of  the  equations  from  7  to  18  in  exercise 
105? 


QUADRATIC  EQUATIONS  287 

Solve  each  of  the  following  fractional  equations : 

3.  ^^^±l=Sx  +  2.  11.    -^4._l-=3. 

x-1  x-2     a;  +  l 

4.  ^^-^==g^.  12.   J:4-^-+      ^      =0. 

ic  +  1        2  2     3  +  a;     2  +  3ic 

.     3x-8      5a;-2  ,-        5  10  2 


ic  — 2         x-\-5  x-\-l     ic+lO     3ic— 3 

5  a;  —  7  _    a;  —  5  ^^        1         4  1 


7a;-5      2x-13  3-a;     5     9-2a; 


X— 1      a;4-2      x  l-{-x     S  —  x     35 


a;  —  2      X      x-\-(i  x  — 4      a;  — 3 

x—  1         a;  6  x  +  1        X—  1 

10.   ^±1+^+2^29.  jg    l  +  _j_  +  ^_  =  0. 

j;  +  2     a;  +  l     10  3     x  +  3     x  +  U 

19.   _i^+     4  15 


20. 


x—\     x  —  Z      a;  +  3 

2  a!  -  3  ^  a;  -  5      5  a;  -  16 
4  12  a;-l   ' 


^,     2a;-2^3-3a;  5 


2a;-3     3a;-2     8a;- 12 

22.  -^ 6_^9^_() 

a;- 3     a;  +  2      3-a; 

23.  2a:-l  ^  13  ^  3  a;  +  5 


2a;  +  l      11      3  a; -5 

3.r 4  4     _ 

.«  -  2      a;  +  3      2  -  a;  ■ 


288  ELEMENTS   OF  ALGEBRA 


AlO.        —  -j- 


26. 

27. 


a;  +  l      x—1      x^  —  1 

1 1      ^  13 

x^-3x     d-x^     16  aj* 

1  17  1 


a^-1      1-x     8      a;-f-l 


28.    ^^+       1 


29.    -^-+        ^^  ^ 


3x-6      72(aj  +  2)      x"-^ 


07^  —  4      2  —  ic  'dx-\-^ 

X         g;  4- 1  _  a;  —  2      a;  —  1 
*a.'  +  l      x-\-2     X  —  1         X 

32.    1  +  ^=1      +      1 


xx-\-4:X-{-lx-\-2 

33         ^     _  ^  —  3  ,       ^     _  ^  +  ^^  ^  2^ 
i»  —  3  ic         aj  +  3         a?         3 

34.  ^  +  l  =  a  +  l.  38.    5  +  «  =  ^  +  «. 

ax  a     X     a     b 

35.  ^L_  +  ^  =  4.  39.    1 =  1_1_1. 

a  +  a;      a  —  x  x  —  a  —  b      x     a      b 

36.  ^  +  ^  =  1  +  1.        40.    ^  +  ^  =  a  +  .. 

a?  —  a     a?  —  6      a     b  x  —  b     x  —  a 

37.  _^+   A_  =  «  +  ft.        41.    ^^ =  i  +  i  +  i- 

a?  —  o      X  —  b      b      a  x  -\-  a  -\-  b      x     a      b 

42.    ^_  +  ^_=_i-+      1 


a;  +  a     a;  +  6      c  +  a      c  +  b 


43.    _^_  +  -^_  =  _^_+     ^ 


X -]- a     X -\- b     c  +  a     c-\-b 


QUADRATIC  EQUATIONS  289 


..     X  -{-  a  ,  X  -{-b     X  —  a  ,  X  —  h 
44. \ — — 1 ^. 

x  —  a     x—o     x^a     x -\- o 

a      ,      b             a  —  c      ,       b  -\-  c 
45. 1 -— =  — ; 1- 


46. 


x-\-a     x  +  b     x-\-a  —  c     jc  +  6 -f- c 
1  1 


x-i-a-\ x  —  a-\- 


x-\-b  x  —  b 

^^     a-{-b      a  +  c  ^  2(a-\-b  +  c) 
X  -{-  b     X  -\-  c        x^-\-  b  -\-  c 

296.  The  following  examples  illustrate  how  any  quadratic 
expression  can  be  factored  by  writing  it  as  the  difference  of 
two  squares. 

Ex.1.  a;2  +  4x  +  0  =  a;2  +  4x  +  4 -(-5) 
=  (x  +  2)2-(V:^)2 
=  (^x  +  2  +^/^=rl)ix  +  2  -V^l). 

Ex.2.   3at2  +  2a;-^  =  [(3x)24.2(3x)-44]--3 
=  [(3a;  +  l)2-45]-3 
=  (3  a;  +  1  +  3\/6) (3  a;  +  1  -  3V6) -4- 3 
=  (x  +  i+V5)(3x  +  l-3  V6). 

Exercise  107. 
Factor  each  of  the  quadratic  expressions: 

1.  x'-^Gx  +  T. 

2.  x^-\-Sx-\-5. 

3.  iB^-lOoj  +  Sl. 

4.  9a^- 6a; -26. 

5.  Sx^-{-6x-3. 

6.  a^-Ux  +  52. 


7. 

ar2  4-10x  +  40. 

8. 

a;2_8a;  +  32. 

9. 

x^-ix-i. 

10. 

a,.2_^_2a;_3. 

11. 

x^-ix-{-5. 

12. 

3ar^_8a:  +  7. 

290  ELEMENTS   OF  ALGEBRA 

13.  Put  each  of  the  twelve  foregoing  trinomials  equal  to 
0,  and  determine,  (1)  the  sum  and  the  product  of  the  roots 
of  each  resulting  equation,  (2)  the  character  of  the  roots 
as  real,  imaginary,  or  complex. 

Factor  each  of  the  following  expressions  and  then  find 
the  roots  of  the  equation  formed  by  putting  it  equal  to  0 : 

14.  a^-4a;  +  16.  17.    4  x^ -|- 8  a;  +  10. 

15.  x'-Qx^ll.  18.    9a^  +  18a:-|-18. 

16.  a^-8a;  +  20.  *  19.    16 a^  +  32 a;  +  27. 


CHAPTER   XX 

PROBLEMS 

297.  The  solving  of  a  problem  by  equations  consists  of 
three  distinct  parts : 

(i)  The  statement  of  the  conditions  of  the  problem  by  one 
or  more  equations. 

(ii)   The  solving  of  these  equations. 

(iii)  The  discussion.  A  problem  may  require  for  an 
answer  a  wliole  number,  an  arithmetic  number,  a  real  num- 
ber, or  numbers  having  some  relation  that  is  not  expressed 
by  the  equations. 

To  state  these  and  other  such  conditions  of  a  problem, 
and  to  determine  what  solutions  of  the  equations  give 
answers  to  the  problem,  is  called  the  discussion  of  the 
problem. 

Prob.  1.  Eleven  times  the  number  of  persons  in  a  room  is  equal  to 
twice  the  square  of  that  number  increased  by  12.  How  many  pei*sons 
are  in  the  room  ? 

Statement.     Let  x  =  the  number  of  persons  ; 

then  lla;  =  2x2  +  12.  (1) 

Solving  (1),  we  obtain  a;  =  4,  x  =  3/2.  (2) 

Equations  (2)  are  together  equivalent  to  (1). 

Discussion.  The  number  of  persons  must  be  an  arithmetic  xchole 
number  which  satisfies  one  of  the  equations  in  (2)  ;  but  4  is  the  only 
such  number.     Hence  the  one,  and  only,  answer  is  4  persons. 

291 


292  ELEMENTS   OF  ALGEBRA 

Prob.  2.  A  train  travels  300  miles  at  a  uniform  rate ;  if  the  rate 
had  been  5  miles  an  hour  more,  the  journey  would  have  taken  2  hours 
less.     Find  the  rate  of  the  train. 

Statement.     Let  x  =  the  number  of  miles  travelled  per  hour ; 
then  300  -^  x  =  the  number  of  hours  required  for  the  journey, 

and        300  h-  (ic  +  5)  =  the  number  of  hours  the  journey  would  have 
taken  if  the  rate  had  been  increased  5  miles 
an  hour. 
Hence,  by  the  conditions  of  the  problem,  we  have 

300  ^   300       2  m 

X       X  +  5       ' 

Solving  (1),  we  obtain  x  =  25,  x  =  —  30.  (2) 

Discussion.     The  number  of  miles  per  hour  must  be  an  arithmetic 

number  which  satisfies  one  of  the  equations  in  (2)  ;  but  25  is  the  only 

such  number.     Hence  the  one  and  only  answer  is  25  miles  an  hour. 

Prob.  3.  The  square  of  the  number  of  dollars  a  man  is  worth  exceeds 
by  300  twenty  times  that  number.     How  much  is  the  man  worth  ? 

Statement.     Let  x  =  the  number  of  dollars  the  man  is  worth  ; 
then  x2  =  20  X  +  300.  (1) 

Solving  (1),  we  obtain  x  =  30,  x  =  —  10. 

Discussion.  If  a  debt  is  regarded  as  a  negative  possession,  both  of 
these  roots  give  answers  ;  that  is,  the  man  either  has  $  30  or  owes  $  10. 

Prob.  4.  The  sum  of  the  ages  of  a  father  and  son  is  100  years ;  and 
one-tenth  of  the  product  of  their  ages,  in  years,  exceeds  the  father's 
age  by  180.     How  old  is  each  ? 

Statement.     Let  x  =  the  number  of  years  in  the  father's  age  ; 
then  100  —x  =  the  number  of  years  in  the  son's  age. 

Hence  0.1  a:(100  -  a;)=  x  +  180.  (1) 

Solving  (1),  we  obtain  x  —  60,  and  100  —  x  =  40, 

or  X  =  30,  and  100  -  x  =  70. 

Discussion.  The  father  must  be  older  than  the  son ;  hence  the 
father  must  be  60,  and  the  son  40,  years  old. 

Both  of  the  solutions  of  (1)  would  give  answers  if  the  jjroblem  read 
as  follows  :  The  sum  of  the  ages  of  tioo  persons  is  100  years ;  and  one- 
tenth  of  the  product  of  their  ages,  in  years,  exceeds  the  age  of  one  of 
them  by  180.     How  old  is  each  ? 


PROBLEMS  293 

Prob.  5.  Find  a  real  number  whose  square  increased  by  13  is  equal 
to  4  times  the  number. 

Statement.     Let  x  =  the  number  ; 

then  a;2  +  13=4a;.  (1) 

Solving  (1),  we  obtain     x  =  2±  SV^.  (2) 

Discussion.  Since  there  is  no  real  number  which  satisfies  (2),  the 
problem  is  impossible. 

If  the  word  '■'•reaV  were  omitted  in  the  problem,  both  the  values 
of  X  in  (2)  would  be  answers. 

Prob.  6.  A  cistern  can  be  filled  by  two  pipes  running  together  in 
22^  minutes  ;  the  larger  pipe  alone  would  fill  the  cistern  in  24  minutes 
less  than  the  smaller  one.     Find  in  what  time  each  would  fill  it. 

Statement.  Suppose  the  larger  pipe  to  fill  the  cistern  in  x  minutes  ; 
then  the  smaller  pipe  will  fill  it  in  x  +  24  minutes.  Also,  1  /x  and 
l/(x  +  24)  are  the  portions  of  the  cistern  which  each  pipe  will  fill  in 
one  minute,  and  1  /22  J  is  the  portion  that  both  together  will  fill  in  one 
minute. 

Hence  -  +  — —  =  — •  (1) 

X     a;  +  24     22^  ^  ^ 

Solving  (1),  we  obtain  x  =  S6,  x  =  — 15.  (2) 

Discussion.  The  answer  must  be  an  arithmetic  number,  but  36  is 
the  only  such  number  which  will  satisfy  either  equation  in  (2). 

Hence  the  larger  pipe  would  fill  the  cistern  in  36  minutes,  and  the 
smallei*  one  in  36  +  24,  or  60,  minutes. 


Exercise  108. 

1.  Find  two  arithmetic  numbers  one  of  which  is  4  times 
the  other,  and  whose  product  is  196. 

2.  Find  two  arithmetic  numbers  whose  sum  is  25,  and 
whose  product  is  144. 

3.  Find  two  numbers  whose  sum  is  15,  and  whose  pro- 
duct is  -  250. 

4.  Divide  71  into  two  parts,  the  sum  of  the  squares  of 
which  is  2561. 


294  ELEMENTS   OF  ALGEBRA 

5.  A  rectangular  court  is  5  yards  longer  than  it  is 
broad;  its  area  is  1886  square  yards.  Find  its  length 
and  breadth. 

6.  The  sum  of  the  squares  of  two  consecutive  whole 
numbers  is  1013.     Find  the  numbers. 

7.  The  sum  of  the  reciprocals  of  two  consecutive  whole 
numbers  is  Jf .     Find  the  numbers. 

8.  If  a  train  travelled  5  miles  an  hour  faster,  it  would 
take  1  hour  less  to  travel  210  miles.  Find  the  rate  of  the 
train. 

9.  The  perimeter  of  a  rectangular  field  is  500  yards,  and 
its  area  is  14,400  square  yards.  Find  the  length  of  the 
sides. 

10.  The  perimeter  of  one  square  exceeds  that  of  another 
by  100  feet;  and  the  area  of  the  larger  square  exceeds  3 
times  the  area  of  the  smaller  by  325  square  feet.  Find  the 
length  of  their  sides. 

11.  A  lawn  50  feet  long  and  34  feet  broad  has  a  path  of 
uniform  width  round  it ;  the  area  of  the  path  is  540  square 
feet.     Find  its  width. 

12.  A  man  travels  108  miles,  and  finds  that  he  could 
have  made  the  journey  in  4^  hours  less  had  he  travelled  2 
miles  an  hour  faster.     At  what  rate  did  he  travel  ? 

13.  The  product  of  the  sum  and  difference  of  an  arith- 
metic number  and  its  reciprocal  is  3|.     Find  the  number. 

14.  A  cistern  can  be  filled  by  2  pipes  in  33 J  minutes. 
To  fill  the  cistern,  the  larger  pipe  takes  15  minutes  less 
than  the  smaller.  Find  in  what  time  it  will  be  filled  by 
each  pipe  singly. 

15.  A  hall  can  be  paved  with  200  square  tiles  of  a  certain 
size;  if  each  tile  were  one  inch  longer  each  way,  it  would 
take  128  tiles.     Find  the  size  of  the  tile. 


PROBLEMS  295 

16.  There  are  two  square  buildings  paved  with  stones 
each  a  foot  square.  The  side  of  one  building  exceeds  that 
of  the  other  by  12  feet,  and  the  two  pavements  together 
contain  2120  stones.     Find  the  sides  of  the  buildings. 

17.  Find  the  number  such  that  the  product  of  the  num- 
bers obtained  by  adding  to  it  3  and  5  respectively  is  less  by 
1  than  the  square  of  its  double. 

18.  The  plate  of  a  mirror  is  18  inches  by  12,  and  it  is 
to  be  framed  with  a  frame  of  uniform  width,  whose  area  is 
to  be  equal  to  that  of  the  glass.  Find  the  width  of  the 
frame. 

19.  A  and  B  distribute  $  100  each  in  charity ;  A  relieves 
5  persons  more  than  B,  and  B  gives  to  each  $  1  more  than 
A.     How  many  did  they  each  relieve  ? 

20.  The  difference  between  the  hypotenuse  and  two  sides 
of  a  right-angled  triangle  is  3  and  6  respectively.  Find  the 
sides. 

21.  In  the  centre  of  a  square  garden  is  a  square  lawn; 
outside  this  is  a  gravel  walk  4  feet  wide,  and  then  a  flower 
border  6  feet  wide.  If  the  flower  border  and  lawn  together 
contain  721  square  feet,  find  the  area  of  the  lawn. 

22.  What  is  the  property  of  a  person  whose  income  is 
$  2150,  when  he  has  |  of  it  invested  at  4  per  cent,  ^  at  3 
per  cent,  and  the  remainder  at  2  per  cent  ? 

23.  A  person  bought  a  certain  number  of  oxen  for  $  1200, 
and,  after  losing  3,  sold  the  rest  for  $  20  a  head  more  than 
they  cost  him,  thus  gaining  $  35  by  the  bargain.  How 
many  oxen  did  he  buy  ? 

24.  A  can  do  a  piece  of  work  in  10  days ;  but  after  he 
has  been  upon  it  4  days,  B  is  sent  to  help  him,  and  they 
finish  it  together  in  2  days.  In  what  time  would  B  have 
done  the  whole  work  ? 


296  ELEMENTS   OF  ALGEBRA 

,25.  A  and  B  can  reap  a  field  together  in  7  days^  which  A 
alone  could  reap  in  10  days.  In  what  time  could  B  alone 
reap  it  ? 

26.  A  can  build  a  wall  in  8  days,  which  A  and  B  can  do 
together  in  5  days.     How  long  would  B  take  to  do  it  alone  ? 

27.  A  does  f  of  a  piece  of  work  in  10  days,  when  B 
comes  to  help  him,  and  they  take  3  days  more  to  finish  it. 
How  long  would  B  take  to  do  it  alone  ? 

28.  The  tens'  digit  of  a  certain  number  exceeds  the  units' 
digit  by  4,  and  when  the  number  is  divided  by  the  sum  of 
the  digits,  the  quotient  is  7.     Find  the  number. 

29.  Find  a  number  of  three  digits,  each  greater  by  1  than 
that  which  follows  it,  so  that  its  excess  above  ^  of  the  num- 
ber formed  by  inverting  the  digits  shall  be  36  times  the 
sum  of  the  digits. 

30.  A  detachment  from  an  army  was  marching  in  regular 
column,  with  5  men  more  in  depth  than  in  front ;  but  on 
the  enemy  coming  in  sight,  the  front  was  increased  by  845 
men,  and  the  whole  was  thus  drawn  up  in  5  lines.  Find 
.the  number  of  men. 

31.  The  sum  of  two  numbers  is  14,  and  the  quotient  of 
the  less  divided  by  the  greater  is  -^^  of  the  quotient  of  the 
greater  divided  by  the  less. 

32.  Find  two  fractions  whose  sum  is  f,  and  whose  differ- 
ence is  equal  to  their  product. 

33.  Two  men  start  at  the  same  time  to  meet  each  other 
from  towns  which  are  25  miles  apart.  One  takes  18  minutes 
longer  than  the  other  to  walk  a  mile,  and  they  meet  in  5 
hours.     How  fast  does  each  walk  ? 

Let  X  =  the  number  of  minutes  it  takes  the  first  man  to  walk  a  mile. 

34.  A  and  B  together  can  do  a  piece  of  work  in  a  certain 
time.     If  they  each  did  one-half  of  the  work  separately,  A 


PROBLEMS  297 

would  have  to  work  one  day  less,  and  B  2  days  more  than 
before.  Find  the  time  in  which  A  and  B  together  do  the 
work. 

35.  A  man  bought  a  certain  number  of  railway  shares 
for  $  1875 ;  he  sold  all  but  15  of  them  for  $  1740,  gaining 
$  4  per  share  on  their  cost  price.  How  many  shares  did 
he  buy  ? 

36.  The  denominator  of  a  fraction  exceeds  the  numerator 
by  4 ;  and  if  5  is  taken  from  each,  the  sum  of  the  reciprocal 
of  the  new  fraction  and  4  times  the  original  fraction  is  5. 
Find  the  original  fraction. 

37.  A  person  swimming  in  a  stream  which  runs  Ij  miles 
per  hour  finds  that  it  takes  him  4  times  as  long  to  swim  a 
mile  up  the  stream  as  it  does  to  swim  the  same  distance 
down.    At  what  rate  does  he  swim  ? 

38.  AVhat  is  the  property  of  a  person  whose  income  is 
$  1140,  when  one-twelfth  of  it  is  invested  at  2  per  cent, 
one-half  at  3  per  cent,  one-third  at  4.^  per  cent,  and  the 
remainder  pays  him  no  dividend  ? 

39.  A  person  having  7  miles  to  walk  increases  his  speed 
one  mile  an  hour  after  the  first  mile,  and  is  half  an  hour 
less  on  the  road  than  he  would  have  been  had  he  not  altered 
his  rate.     How  long  did  it  take  to  walk  the  7  miles  ? 

Let  X  miles  an  hour  be  his  rate  at  first. 

40.  The  diagonal  and  the  longer  side  of  a  rectangle  are 
together  5  times  the  shorter  side,  and  the  longer  side  exceeds 
the  shorter  by  35  yards.     Find  the  area  of  the  rectangle. 

41.  The  price  of  photographs  is  raised  50  cents  per 
dozen ;  and,  in  consequence,  4  less  than  before  are  sold  for 
$  5.     Find  the  original  price. 

42.  A  boat's  crew  can  row  8  miles  an  hour  in  still  water. 
What  is  the  speed  of  a  river's  current  if  it  takes  them  2 


298  ELEMENTS   OF  ALGEBRA 

hours  and  40  minutes  to  row  8  miles  up  and  8  miles  down 
the  river  ? 

Let  X  =  the  number  of  miles  the  current  runs  in  an  hour  ; 


then  — ^^—  +  —2—  =  -. 

8  +  X     8  -  X     3 

43.  At  a  concert  $  300  was  received  for  reserved  seats, 
and  the  same  amount  for  unreserved  seats.  A  reserved  seat 
cost  75  cents  more  than  an  unreserved  seat,  but  360  more 
tickets  were  sold  for  unreserved  than  for  reserved  seats. 
How  many  tickets  were  sold  all  together  ? 

44.  Out  of  a  cask  containing  GO  gallons  of  alcohol  a 
certain  quantity  is  drawn  off  and  replaced  by  water.  Of 
the  mixture  a  second  quantity,  14  gallons  more  than  the 
first,  is  drawn  off  and  replaced  by  water.  The  cask  then 
contains  as  much  water  as  alcohol.  How  much  was  drawn 
off  the  first  time  ? 

Let  X  —  the  number  of  gallons  drawn  off  the  first  time  ;  then,  in 

the  first  mixture, 

60  —  X  =  the  number  of  gallons  of  alcohol, 

and  X  =  the  number  of  gallons  of  water. 

60  -  X  xgQ  _         -^4>^^  ^ ^gQ  _  ^  _  ^4^^_^  ^  _^  14. 
60     ^  ^60 

45.  A  cyclist  rode  180  miles  at  a  uniform  rate.  If  he 
had  ridden  3  miles  an  hour  slower  than  he  did,  it  would 
have  taken  him  3  hours  longer.  How  many  miles  an  hour 
did  he  ride  ? 

46.  A  man  drives  to  a  certain  place  at  the  rate  of  8  miles 
an  hour.  Returning  by  a  road  3  miles  longer  at  the  rate  of 
9  miles  an  hour,  he  takes  7|  minutes  longer  than  in  going. 
How  long  is  each  road  ? 

47.  A  father's  age  is  equal  to  the  united  ages  of  his 
5  children,  and  5  years  ago  his  age  was  double  their  united 
ages.     How  old  is  the  father  ? 


PROBLEMS  299 

48.  A  and  B  are  two  stations  300  miles  apart.  Two 
trains  start  simultaneously  from  A  and  B,  each  to  the  oppo- 
site station.  The  train  from  A  reaches  B  9  hours,  the  train 
from  B  reaches  A  4  hours,  after  they  met.  When  did  they 
meet,  and  what  was  the  rate  of  each  train  ? 

49.  If  a  carriage  wheel  14|  feet  in  circumference  takes 
one  second  more  to  revolve,  the  rate  of  the  carriage  per  hour 
will  be  2J  miles  less.     How  fast  is  the  carriage  travelling  ? 

Let  X  =  number  of  miles  travelled  per  hour  ;  then 


X     10      X  -  2f 

50.  The  number  of  square  inches  in  the  surface  of  a 
cubical  block  exceeds  the  number  of  inches  in  the  sum  of 
its  edges  by  288.     Find  its  edge  and  volume. 

51.  A  cistern  can  be  filled  by  2  pipes  running  together 
in  2  hours  55  minutes.  The  larger  pipe  by  itself  will  till  it 
sooner  than  the  smaller  one  by  2  hours.  Find  the  time  in 
which  each  pipe  separately  will  fill  it. 

52.  My  gross  income  is  $  3000.  After  paying  the  income 
tax,  and  then  deducting  from  the  remainder  a  percentage 
less  by  1  than  that  of  the  income  tax,  the  income  is  reduced 
to  $  2736.     Find  the  rate  per  cent  of  the  income  tax. 

53.  A  set  out  from  C  toward  D  at  the  rate  of  5  miles 
an  hour.  After  he  had  gone  45  miles,  B  set  out  from  D 
toward  C,  and  went  every  hour  -^^  of  the  entire  distance. 
After  travelling  as  many  hours  as  he  went  miles  in  an  hour, 
he  met  A.     Find  the  distance  from  C  to  D. 


CHAPTER  XXI 
IRRATIONAL  EQUATIONS 

298.  An  irrational  equation  is  an  equation  one  or  both  of 
whose  members  is  irrational  in  an  unknown. 

In  this  chapter,  as  heretofore,  the  radical  sign  will  denote 
only  the  principal  root  of  a  number  or  expression. 


E.g.^  Vx^  —  2  =  X  —  7  is  an  irrational  equation,  and  Vx^  -  2  de- 
notes only  the  principal  square  root.  Note  that  we  cannot  speak  of 
the  degree  of  this  or  any  other  irrational  equation. 

In  solving  irrational  equations  we  use  the  following 
principle ; 

299.  If  both  members  of  an  irrational  equation  are  raised 
to  the  same  integral  power,  the  derived  equation  will  have  all 
the  roots  of  the  given  one  and  often  others  in  addition. 

Proof     Let  A  =  B  (1) 

be  the  given  irrational  equation. 

Squaring,  A^  =  B\  (2) 

By  §§  105  and  106,  (2)  is  equivalent  to  the  equation 

{A-B){A  +  B)  =  (>.  (3) 

By  §  74,  the  roots  of  (3)  include  those  of  A  —  B=0,  or 
(1)  ;  hence  no  root  is  lost  by  squaring  (1). 

But  the  roots  of  (3)  include  also  those  of  A  +  B  =  0,ov 
A  =  —  B;  hence  any  root  of  A  =  —  B  which  is  not  a  root 
of  ^  =  J5  must  be  introduced  by  squaring  (1). 

In  like  manner  the  principle  can  be  proved  for  any  other 
positive  integral  power. 

300 


IRRATIONAL  EQUATIONS  301 

Ex.  1.   Solve  the  equation      x  —  6  =  —  Vx  —  6.  (1) 

Square,  x2  -  12  x  +  36  =  x  -  6.  (2) 

Transpose,  x^-lSx-{-4t2  =  0. 

Factor,  (x  -  6)  (x  -  7)  =  0.  (3) 

Now  (3) ,  or  (2) ,  is  satisfied  when  x  =  6  and  x  =  7  ;  but  (1)  is  satisfied 
only  when  x  =  6.     Hence  by  squaring  (1)  the  root  7  was  introduced. 
By  §  299,  the  roots  of  (2)  include  those  of  (1)  and  also  those  of 


A  =  -  B,  or  X  -  6  =  Vx-e.  (4) 

Equation  (4)  is  satisfied  both  when  x  =  6  and  when  x  =  7. 

Hence  if  it  had  been  required  to  solve  (4),  by  squaring  we  would 
have  obtained  (2),  and  no  root  would  have  been  introduced. 

Notice  that  we  cannot  say  that  (2)  is  equivalent  to  (1)  and  (4) 
jointly  (as  would  be  the  case,  by  §  290,  were  (1)  and  (4)  rational 
equations);  for  (2)  has  only  two  roots,  while  (1)  and  (4)  together  have 
three  roots,  6  being  a  root  of  each. 

Observe  that,  since  we  cannot  speak  of  the  degree  of  an  irrational 
equation,  we  do  not  know  how  many  roots  it  has  until  we  have  solved  it. 


Ex.  2.    Solve  2  -  V2  X  +  8  +  2  Vx  +  6  =  0.  (1) 

Our  purpose  being  to  obtain  a  rational  equation,  it  is  better  before 

squaring  to  put  the  more  complex  surd  in  one  member  by  itself,  as 

below. 

Transpose,  2  +  2  Vx  +  6  =  a/2  x  +  8. 


Square,  4  4-  8 Vx  +  6  +  4x  +  20  =  2x  +  8. 


Transpose,  x  +  8  =  -  4Vx  +  5.  (2) 

Square,  x2  +  16  x  +  64  =  16  x  +  80. 

Transpose,  x^  -  16  =  0.  (3) 

Hence,  by  §  299,  if  (1)  has  any  root,  it  is  4  or  —  4.  But  neither 
X  =  4  nor  x  =  —  4  satisfies  (1)  ;  hence  (1)  has  no  root,  i.e.,  it  is  impos- 
sible, and  therefore  both  roots  of  (3)  were  introduced  by  squaring  (1) 
and  (2). 

If  we  use  both  the  positive  and  the  negative  values  of  V2x  +  8  and 


vx  +  6,  we  obtain  in  addition  to  (1)  the  three  equations. 


2-V2x4-8-2Vx  +  5  =  0,  (4) 


2+V2x  +  8-2\/x+5  =  0,  (5) 


2+V2x  +  8  +  2Vx-f5=0.  (6) 


302  ELEMENTS   OF  ALGEBRA 

By  treating  (4),  (5),  or  (6)  as  we  did  (1),  we  would  obtain  (3). 
Hence  the  roots  of  (3)  include  the  roots  of  each  of  the  four  equations, 
(1),  (4),  (5),  and  (6). 

By  trial  we  find  that  —  4  is  a  root  of  (4),  —  4  and  +  4  are  both 
roots  of  (5),  and  neither  —  4  nor  +  4  is  a  root  of  (6)  ;  that  is,  (6) 
expresses  an  impossible  condition  as  well  as  (1). 

Hence  in  rationalizing  (1)  or  (6),  i.e.,  in  deriving  the  rational 
equation  (3)  from  (1)  or  (6),  two  roots  are  introduced. 

In  rationalizing  (4),  one  root  is  introduced. 

In  rationalizing  (5),  no  root  is  introduced. 

Exercise  109.  i^ 

Solve  each  of  the  following  irrational  equations : 


1.    Vaj  — 5  =  3.  7.    Va;4-25  =  1+V^. 


2.  7  -  Va;  -  4  =  3.  8.    Vi»  +  3  -f  ^x  =  5. 

3.  V5a;-l=2Vaj  +  3.  9.    V8a; +  55  -  3  =  2V2^. 

4.  2V3-7a;=3V8fl7-12.      10.    10  -V25 -\-'i)  x  =  3^x. 


5.  .V9x'-llx-o  =  3x-2.     11.    V9ic-8  =  3Va.'-h4-2. 

6.  V4.x^-7 x-^l  =  2x-l^.     12.    ■Vx-4.-\-3=Vx-\-ll. 

13.  In  each  of  the  foregoing  examples,  from  what  other 
irrational  equation  or  equations  would  we  have  derived  the 
same  rational  equation  ? 

14.  ■\/Sx-{-17  -V2x  =  -V2x-\-9. 

15.  V3a;-ll+V3^=Vl2x-23. 


16.  -Vi2x-5+VSx-l=^27x--2. 

17.  Vx  +  3  +  Va;  +  8  =  V4  a;  4-  21. 


18.    Va5  +  2+V4a7  +  l=V9aj  +  7. 


19.    ■\/x  -i- 4:  ab  =  2  a -\-  ^x. 


20.    ■^x+V4:a-^x=2Vb  +  x. 


21.    Vx  —  l-\-^x  =  2^^x. 


IRRATIONAL  EQUATIONS  303 


22.    ViC  +  o  +  V^  =  10  -=-  Y^ic. 


23.  V^- vaJ-8  =  2H-V.T-8. 

24.  Vl  +  a;  +  V^  =2  -J-  VI  +  a;. 


25.  2 VaJ  —  V4ic— 3  =  l^V4a;  —  3. 

26.  V^  -  '^  =  1  -^  (V^  +  ")• 

27.  -^Hi- =  3  +  ^^  +  1. 

V«  — 1  2 

Simplify  the  first  member  in  example  27. 


28. 

^ 

-  2Va;     2 

-1 

2  = 

-2  +  1 

29. 

V2  +  «  +  V2  -  a; 

V2  +  a;-V2-a! 

30. 

1 

1- 

+       1       +       ^ 

a;      ^x  -h  1      V^  ~ 

-  =  0. 

In  the  next  seven  examples,  first  reduce  the 

improper  fractions  to 

lixed  expressions : 

31. 

4-  3  _  S^x  -  5 
-  2     3  V-x-  -  13 

32. 

9V^ 
3v 

r-23_6V«-17 

'X  -  8       2  V-»  -  <3 

V^  +  -    V-'«  +  '^ 

34     6Va?-7  g^7Vx-26 

V^-l  7V.T-21 

35     2Va;-l^Vx-2  3^     12v^-ll  ^  6^^4-5 

*   Va'  +  t     V^-i  '    4va.'-4f      2V^  +  t' 


36. 


304  ELEMENTS  OF  ALGEBRA 

39.  ^(a-x)+^(b-x)=^(a-\-b-2x). 

40.  ^(ax  +  b^)  —  -^(bx  i- a^)  =  a  —  b. 

41.  ^(a-{-x)-\--^(b  +  x)=^(a  +  b-^2x). 

42.  ^(a-x)+^(b~-x)=-y/(2a-{-2b). 

300.  Equations  in  quadratic  form.  If  an  equation  has  only 
two  unknown  terms,  and  if  the  unknown  factor  of  one  of 
these  terms  is  the  square  of  the  unknown  factor  of  the 
other,  the  equation  is  in  quadratic  form. 


E.g.,  since  x^  +  Sx  is  the  square  of  vx^  +  3x,  the  equation 


(x2  +  3x)+  5vxM-3x  =  7  is  in  quadratic  form. 

The  following  examples  illustrate  how  the  principles  of 
quadratic  equations  can  be  applied  to  irrational  equations 
which  are,  or  can  be  put,  in  quadratic  form. 


Ex.1.    Solve  2x2  +  3x-5\/2x'^  +  3x  +  9=-3.  (1) 


Add  9,  (2x2  +  3x  +  9)-5V2x2  +  3x  +  9  =  6.  (2) 


Since  2  x^  +  3  x  +  9  is  the  square  of  V2  x^  +  3  x  +  9,  equation  (2) 
is  in  quadratic  form.    Transposing  6  and  factoring,  we  have 


(\/2  x2  +  3  X  +  9  -  6)  (  V2  x2  +  3  X  +  9  +  1)  =  0.  (3) 

The  roots  of  (3)  include  the  roots  of 

V2  x2  +  3  X  +  9  =  6,  (4) 

and  of  \/2  x2  +  3  X  +  9  =  -  1,  (5) 

but  no  others. 

The  roots  of  (4)  are  3  and  —  4J ;  while  (5)  is  an  impossible  equa- 
tion, since  a  principal  square  root  cannot  be  a  negative  number. 

What  would  be  the  roots  of  (1),  if  the  sign  before  the  radical 
were  +  ? 


Ex.  2.    Solve  3  x2  -  7  +  3  V3  x2  -  16  X  +  21  =  16  X.  (1) 

Transposing  16  x  and  adding  28  —  28,  we  obtain 


(3x2  -  16x  +  21)  +  3V3x2  -  16x  +  21  -  28  =  0.  (2) 


IRRATIONAL   EQUATIONS  305 


Factor,  (VSx^-  16  a: +  21  -  4)  ( V3  x^  -  16  x  +  21  +  7)  =  0.     (3) 
The  roots  of  (3)  include  the  roots  of 


V3  x2  -  16  a;  +  21  =  4,  (4) 

and  of  V3  x2  -  16  X  +  21  =  -  7,  (5) 

but  no  others. 

The  roots  of  (4)  are  5  and  1/3,  and  (5)  is  impossible. 

What  would  be  the  roots  of  (1),  if  the  sign  before  the  radical 
were  —  ? 

If  we  could  not  factor  (2)  by  inspection,  by  §  293  we  would  have 


V3x2-16x  +  21  =  -  f  ±  Vf  +  28  =  +  4  or  -  7. 

Exercise  110. 
Solve  each  of  the  following  irrational  equations : 


1. 

Sx^-4:X-\-V3x'-4:X-6  =  lS. 

2. 

a^  _  a; -f  4  4- Vur  -  a? -h  4  =  2. 

3. 

a^  +  2a;  -  Var*  H- 2a;  -  6  =  12. 

4 

1  1   1  Var*  \  X  \  5  — 

Va;2  +  a;  -f  5 

5. 

a^  4.  V4ar^  H-  24 a;  =  24  -  6 a;. 

6.    2a^-\-6x  =  l-Va^-]-3x  +  l. 


7.    2  (2  a;  -  3)  (a;  -  4)  -  V2  ar^  -  11  a;  +  15  =  60. 


8.    ^4.0^ -{-2  X -^  7  =  12  x'-^Gx- 119. 


9.    2a;2_2a;_17  +  2V2a;2_3^_l_7^^^ 


10.  3a;(3-a;)  =  ll-4Var'-3a;  +  5. 

11.  2ar2-4a;-Va;^-2a;-3  =  9. 


CHAPTER   XXII 

HIGHER  EQUATIONS 

301.  The  following  examples  illustrate  how  the  princi- 
ples of  quadratic  equations  are  applied  to  higher  equations 
which  are,  or  can  be  put,  in  quadratic  form. 

Ex.  1.    Solve     (x2  +  2a;)2-5(a;2  +  2a;)-14  =  0.  (1) 

Factor,  (a:2  +  2  x  -  7)(a;2  +  2  a;  +  2)  =  0.  -  (2) 

Equation  (2)  is  equivalent  to  the  two  equations 

x2  +  2x-7  =  0,     x2  +  2x  +  2  =  0, 
each  of  which  is  readily  solved. 

Ex.  2.    Solve  x*  -  8  x3  +  10  ic2  +  24  X  +  5  =  0.  (1) 

Adding  C  x2  —  6  x^  to  the  first  member,  we  have 

(x*  -  8  a:3  +  16  a;2)  -  6  x2  +  24  X  +  5  =  0, 
or  (a:2-4x)2-6(x2-4x)  +  5  =  0.  (2) 

Factor,  (a;2  _  4  x  -  5)  (x2  _  4  x  -  1)  =  0.  (3) 

Equation  (3)  is  equivalent  to  the  two  equations 

ic2-4x-5  =  0,     x2-4x-l=0, 
whose  roots  are  5,  —  1,  2  i  VS. 

Ex.  3.    Solve  -^^  +  ^^  =  ^.  (1) 

X  —  1  x2  4 

Here  the  second  term  is  the  reciprocal  of  the  first. 
Putting  y  for  the  first  term,  and  therefore  the  reciprocal  of  y  for 
the  second,  (1)  becomes 

y      4 
306 


HIGHER   EQUATIONS  307 


Multiply  by  4  y,  4  y^  -  17  ?/  +  4  =  0. 

Factor,  (y  -  4)  (4  2/  -  1)  =  0. 

.-.  2/  =  4,  or  1/4. 
Hence  (1)  is  equivalent  to  the  two  equations 


^^     =  4  and  -^^  =  1.  (2) 


x-l  x-1      4 

The  roots  of  equations  (2)  are  2,  2,  (1  ±  V—  16)/8. 

Exercise  111. 
Solve  the  following  equations  : 

1.  a:^-5ar'  +  4  =  0.  3.    a;*  -  7  ar^  -  18  =  0. 

2.  a;* -10  0^2.^9^  Q  4^    (oF -l)/9 -i-l/a^  =  1. 

5.  a^  +  lOO/ar^  =  29. 

6.  (a^ -h  a!)2  -  22  (ar'  + a-)  =-40.    • 

7.  (ar^  -  a^)- -  8  (a-2  -  .1-)  =  -  12. 

9.  2»2  +  3a;H-l=  30/(2  ar'  +  3  x). 

10.  ar^  +  3a;-20/(ar  +  3a;)  =  8. 

11.  a^  +  a;  +  1  =  42/(a^  +  a;). 

12.  a;^  -  8  ar'^- 12^2  4- 112  a;  =  128. 

13.  aj*  +  2ar'-3ar-4.r-96  =  0. 

14.  a;*-10.t'3-}-30ar'-25a;  +  4  =  0. 

15.  a;^-14ar^-hGla:2_g4^_^20  =  0. 

16.  ^_  +  ^  +  l_o 


17. 


a;  +  l         x" 

X         a.-^+1^5 
X-  +  1  a;         2 


18     -    ar^-f-2  ar^  +  4.r4-1^5 

ar^  +  4a;-fl  ar'4-2  2* 


308  ELEMENTS   OF  ALGEBBA 

302.  A  binomial  equation  is  an  equation  of  the  form  a;"  =  a, 
where  n  is  a  positive  integer. 

The  binomial  quadratic  equation  x^  =  a  has  already  been 
solved.  Certain  binomial  higher  equations  are  readily  solved 
by  previous  principles. 

Ex.  1.    Solve  the  binomial  cubic  equation  x^  —  1  =  0.  (1) 

Factor,  (a;  -  1)  (a;2  +  x  +  1)  =  0.  (2) 

Equation  (2)  is  equivalent  to  the  two  equations 

X  -  1  =  0,  x2  +  X  +  1  =  0.  (3) 

The  solutions  of  equations  (8)  are  1  and  (—  1  ±  v'—  3)/2. 

Hence,  the  cubic  equation  (1)  has  one  real  and  two  complex 
solutions. 

Since  by  (1),  x^  =  +  1,  tlie  cube  of  each  solution  of  (1)  is  equal  to 
+1  ;  that  is,  +1  has  the  three  cube  roots  +1  (— 1+V— 8)/J,  and 
(—1  —  V—  3)/2.     See  example  32,  exercise  103. 

Since  +  27  =(+  1)  x  27,  the  three  cube  roots  of  +27,  or  the  three 
solutions  of  the  cubic  equation  x^=^21,  can  be  obtained  by  multiply- 
ing the  three  cube  roots  of  +1  by  the  cube  root  of  the  arithmetic  num- 
ber 27. 

Thus  the  three  solutions  of  x^  =+27  are  +3  and  3(-  1  ±  V^^)/2. 

Ex.  2.    Solve  the  binomial  biquadratic  equation  x^  —  1  =  0.  (1) 

Factor,  (x^  -  l)(x2  +  1)  =  0.  •  (2) 

The  solutions  of  (2)  are  ±  1  and  ±  V—  1. 
Hence  +1  has  four  fourth  roots,  two  real  and  two  imaginary. 
The  four  solutions  of  x*  =  81,  or  the  four  fourth  roots  of  +81,  are 
±3and±3V^^. 

Ex.  3.    Solve  x5  =  1,  or  x^  -  1  =  0.  (1) 

Factor,  (x  -  1)  (x*  +  x^  +  x^  +  x  +  1)  =  0.  (2) 

One  solution  of  (2)  is  1,  and  the  other  solutions  are  those  of  the 
equation 

x4  -f  x3  +  x2  +  X  +  1  =  0.  (3) 

Divide  by  x^,  a;2  +  x  +  1  +  -  +  ^  =  0. 


HIGHER   EQUj^TIONS  309 


Addl,  a;2  +  2+l+x  +  -  =  l, 


(..i)%(..l)=x. 


.-.  x2+l  =  K-l±V5)x.  (4) 

Solving  the  two  equations  in  (4),  we  obtain  four  solutions,  all  of 
which  are  complex.  Hence  1,  or  any  other  positive  number,  has  five 
fifth  roots,  one  real  and  four  complex. 

Ex.  4.    Solve  x6  =  1^  or  x5  -  1  =  0.  (1) 

Factor,  (x^  -  1)  (x^  +  1)  =  0, 

or  (a;-l)(a;2  +  x+l)(x+l)(x2-x  +  l)  =  0.  (2) 

Equation  (2)  is  equivalent  to  the  four  equations 

X  -  1  =  0,  x'  +  X  +  1  =  0,  X  +  1  =  0,  x^  -  X  +  1  =  0.  (3) 

Solving  equations  (3),  we  obtain  six  solutions,  two  real  and  four 
complex. 

Hence  1,  or  any  oXhQV  positive  number,  has  six  sixth  roots,  two  real 
and  four  complex. 

Ei.  5.    Solve  x^  =  1,  or  x8  -  1  =  0.  (1) 

•Factor,  (x*  +  1)  (x^  +  1 )  (x^  -  1 )  =  0.  (2) 

The  roots  of  (2)  are  ±1,  ±  V-  1,  and  the  roots  of 

X*  +  1  =  0.  (3) 

Add  2  x2  -  2  x2,       x4  +  2  x2  +  1  -  2  x2  =  0. 

Factor,  (^2  +  1  +  Xy/'l)  (x2  +  1  -  Xy/I)  =  0.  (4) 

Equation  (4)  is  equivalent  to  the  two  equations 

X2  +  1   +  Xy/2  =  0, 

and  x2  +  ,l-xV2  =  0, 

each  of  which  has  two  complex  roots. 


310  ELEMENTS   OF  ALGEBRA 

Hence,  any  positive  number  has  eight  eighth  roots,  two  real,  two 
imaginary,  and  four  complex. 

Observe  that  any  root  of  +  1  or  —  1  is  a  quality-unit. 

Exercise  112. 
Solve  each  of  the  following  binomial  equations : 

1.  ar''  +  l  =  0.  5.    a^  +  l  =  0.  9.    a^^  _  ^4  ^  0^ 

2.  a^  +  27  =  0.         6.    ar'  +  32  =  0.  10.    a:* -625  =  0. 

3.  .T^  +  1  =  0.  7.    x^-\-l  =  0.  11.    af- 243  =  0. 

4.  x'-\-16  =  0.         8.    a.'«H-64  =  0.  12.    x«- 729  =  0. 


CHAPTER   XXIII 

SYSTEMS  INVOLVING  QUADRATIC  AND  HIGHER 
EQUATIONS 

303.  As  in  linear  systems,  so  in  any  other  determinate 
system  tliere  must  be  as  many  independent  consistent  equa- 
tions as  there  are  unknowns. 

In  solving  systems  which  involve  quadratic  or  higher 
equations  we  have  frequent  use  for  the  following  principle 
of  equivalent  systems: 


304.    If  M,  N,  P,  Q  denote  any  integral  unknown  expres 
sions,  then  system  (a) 

PxQ 


0,1 


(a) 


is  equivalent  to  the  four  systems  (6),  (c),  (d),  (e). 


M=0, 
P=0. 


M=0,]  ,  ^=0,1  ,  iV^=0,l 

(").  Q^oM^^   p=oM'^  e=o:}« 


Proof  Any  solution  of  system  (a)  must  reduce  the  factor 
M  or  N  (or  both)  to  0,  and  at  the  same  time  must  reduce 
P  or  Q  (or  both)  to  0. 

Now  any  solution  of  system  (a)  which  reduces  M  to  0 
and  P  to  0  is  a  solution  of  system  (6) ;  any  solution  of  (a) 
which  reduces  M  to  0  and  Q  to  0  is  a  solution  of  (c) ;  and 
so  on.  Hence  any  solution  of  system  (a)  is  a  solution  of 
system  (6),  (c),  (d),  or  (e). 

('onversely,  any  solution  of  system  (b)  reduces  M  to  0  and 
P  to  0,  and  therefore  reduces  M  x  iV  to  0  and  P  x  Q  to  0; 

311 


312  ELEMENTS   OF  ALGEBRA 

hence,  any  solution  of  system  (b)  is  a  solution  of  system 
(a),  and  so  on.  Hence,  any  solution  of  system  (6),  (c),  (d), 
or  (e)  is  a  solution  of  system  (a). 

Whence  system  (o)  is  equivalent  to  the  four  systems  (6), 
(c),  (d),  (e). 


Ex.  1.   Solve  thie  system 

a;2  -  a:?/  -  2  ?/2  =  0,  (1) 

3y2_i0y  +  8  =  0.  (2) 

By  §  201,  system  (a)  is  equivalent  to  (&). 

Factor  (1),                  (^^  _  2  ?/)  (x  +  2/)  =  0.  (3) 

Factor  (2),                   (Zy  -  ^){y -2)=Q.  (4) 

By  §  304,  (6)  is  equivalent  to  the  four  linear  systems  (c), 

a;-2?/  =  0,  1      x-2y  =  0,  I         »:  +  ?/  =  0,  1  x  +  ?/  =  0,  ' 

3  2/-4  =  0.  J         ?/-2  =  0.  J      3?/-4  =  0.  J  ?/-2  =  0.  , 


(a) 


(&) 


(^) 


The  solutions  of  the  four  systems  (c)  are  |,  f  ;  4,2;  —  -f,  | ;  —  2,  2 
which  are  therefore  the  four  solutions  of  (a). 

Ex.  2.    Solve  the  system 

x^  +  2xy  +  y'^  =  36,  (1) 


a:2-2a;i/  =  0.  (2) 

System  (a)  is  equivalent  to  system  (6) . 

From(l),  x^y=±A.  ^^H  (5) 

From  (2),  x{x  -2y)=0.  (4)  J 

By  §  304,  (&)  is  equivalent  to  the  four  linear  systems  (c). 

x  +  ?/  =  4,  1         a;  +  ?/  =  4,  1      a;  +  !/  =  -4,  1         x  +  y  =  -4 


4,  1      a;  +  !/  =  -4,  1         a;  +  y  =  - 
0.  J  x  =  0.      J      ic  -  2  ?/  =  0. 


cc  =  0.  J      a;  —  2?/  = 

In  applying  the  principle  of  this  article  to  system  (6),  observe  that 
the  two  equations  in  (3)  are  equivalent  to  the  equation 

{x-\-y-i)(ix  +  y  +  i)=0. 

The  solations  of  (a)  are  therefore  0,  4  ;  f ,  f  ;  0,  —  4  ;  —  |,  —  f . 


SYSTEMS   OF  QUADRATIC  EQUATIONS  313 

Whenever  one  or  each  of  the  equations  of  a  system  can 
be  resolved  into  two  or  more  equivalent  equations,  the  Jlrst 
step  in  solving  the  system  is  to  apply  the  principle  of  this 
article. 

305.   The  two  examples  in  §  304  illustrate  the  theorem : 
A  system  of  two  quadratic  equations  in  two  unknowns  has, 
in  general,  four,  and  only  four,  solutions. 


Exercise  113. 
Solve  each  of  the  following  systems  of  equations : 

1.  (x-2y)(x-l)  =  0,]  6.    (x-\-y){x-y  +  l)  =  0,] 
x-\.y-i  =  0.             1  (x-\-2)(y  +  S)  =  0.         J 

2.  (x-3)(y-2)  =  0,)  7.    (x-hyf  =  16,\ 


a;  +  2/  =  7.  J  (p-yY 

3.  x'-^xy  +  Zy'-^OA  8.    cr  +  2 a^y  +  2/' =  144,  | 
X  -\-y  =  \.                  J  Q(?  —  2  xy  -\- y"^  =  A:.      J 

4.  a-?/ -7  2/ +  3  a;  =  21,1  9.    a?-{-xy  =  x-\-y. 


.1 


x-\-y  =  2.  )  y^  —  2xy  =  Sy  —  6x. 

5.4:a^  —  xy  =  0,]  10.   a^  —  y^  =  x -\- y,  1 

2a?-32/  =  6.  J  ix?-Sxy  =  5x-15y.) 

306.  A  system  of  two  equations,  one  linear  and  the  other 
quadratic,  can  be  solved  by  first  eliminating  one  unknown  by 
substitution. 


Ex.  1.    Solve  the  system 

aj  +  2  y  =  5, 

S!<« 

«2  +  2y2  =  9. 

Solve  (1)  for  x, 

x  =  5-2y. 

(3) 

From  (2)  and  (8),        (5  - 

-2?/)2  +  2?/-^  =  9. 

(6) 

Factor,                        (3  y  - 

-4)(2  2,-4)=0. 

(4). 

314  ELEMENTS   OF  ALGEBRA 

By  §  201,  (a)  is  equivalent  to  the  system,  (3)  and  (4),  or  (6). 
By  §  304,  (b)  is  equivalent  to  the  two  systems  (c)  and  (d). 

X  =  5  —  2y,]  x  =  6  —  2y,] 

■(c)  (d) 

3?/ -4  =  0.  r  '  y-2  =  0.  J  '  ' 

The  solution  of  (c)  is  7/3,  4/3  ;  and  that  of  (d)  is  1,  2. 
After  the  theory  is  clearly  understood,  the  work  after  equation  (4) 
can  be  abridged  as  below  : 

From  (4),  y  =  4/3,  or  2. 

When  y  =  4/3,  from  (3),  x  =  6-  8/3  =  7/3  ; 

When  y  =  2,  from  (3),       a;  =  5  -  4  =  1. 

This  example  illustrates  the  following  theorem : 

A  syste7n  of  one  linear  and  one  quadratic  equation  in  two 
unknowns  has,  in  general,  two,  and  only  two,  solutions. 

Exercise  114. 
Solve  each  of  the  following  systems : 


1.    x-\-y  =  15,]  S.    x  —  y  =  3. 


} 


xy  =  36.       J  iK^  +  19  +  2/-  =  3  xy 

2.  x-\-y  =  bl,\  d.    2x  —  y  =  5, 
xy  =  518.      J  x-\-3y  =  2xy 

3.  Sx-4.y  =  -12,  I  10.    3x-\-2y  =  5,  1 
3ic2  4-  2  2/2  -  2/  =  48.  J                 x' -  ixy -\- 5y' =  2.) 

4.  x-y  =  10,    1  11.    3x'-2xy  =  15,] 
0^  +  2/2  =  58.1  2x-{-3y  =  12.     J 

5.  3x-\-3y  =  10,)  12 
xy  =  1.              J 

6.  2x-5y  =  0,    1  13.    x^  +  3xy-y^  =  23, 
3)2-3  2/2  =  13.1  x  +  2y  =  7. 

7.  2x-\-3y  =  0,  I        14.    x'-{-if=lS5, 


.    x  +  y  =  15,      I 
a^2  -f  2/'  =  125.  J 


SYSTEMS   OF  QUADRATIC  EQUATIONS  815 

15.    2x-7y  =  2o,  1        17.   x-\-y  =  2,  1 

5x'-{-4:xy+Sy-=2S.}  2x  +  3y  =  6xy.} 

le.   3x  —  31  =  5y,  1  18.    x-^2y  =  7,         1 

a^  -\-  5 xy  +  2o  =  y^. }  3y  +  6x  =  5xy. } 

19.  x-y  =  l,  I 
^-f  =  (^/6)xy.) 

20.  x^-2xy=^0,  (1)| 
4a^  + 92/2 ^225.  (2)  I 

Factor  (1),  a;(x  -  2  y)  =  0.  (3) 

System,  (2)  and  (3),  which  by  §  201  is  equivalent  to  (a),  is  equiva- 
lent to  the  two  systems  (6)  and  (c). 

4a;2  4-9?/2  =  225,  1  4^2  +  9y2  =  225,  1 


a;  =  0.      J  a;-2y  =  0. 

21.    a:2-3a:?/  =  0,      1  23.    Q!^-2xy  +  o  =  0, 


5ar'  +  3/  =  48.J  (a;-2/y  =  4.  J 

22.    2ar-3a;.y  =  0,l  24.    a^  +  4/  =  4a^  +  16, 1 

2/2  +  5an/  =  34.  J  a^ 4- .v'  =  5.  J 

307.  If  each  of  two  quadratic  equations  has  onSf  and  only 
one  J  temi  below  tlie  2d  degree^  and  these  two  terms  are  similar; 
the  system  can  be  solved  by  first  eliminating  the  term  below 
the  second  degree  by  addition  or  subtra^ion. 

Ex.  1.   Solve  the  system  x"^  +  xy  +  2  y-  =  44,  (1)  | 

2a;2-xy  +  2/2  =  16.  (2)}^''^ 

Each  equation  in  system  (a)  has  one,  and  only  one,  term  below 
the  2d  degree,  44  and  1(3,  respectively  ;  and  these  terms  are  similar. 
We  proceed  to  eliminate  the  term  below  the  2d  degree. 

Multiply  (1)  by  4,  4^2  +  4  a;.v  +  8  y2  =  ne.  (3) 

Multiply  (2)  by  11 ,  22  x"^  -  11  xy -\- 11  y"- =  176.  (4) 

Subtract  (3)  from  (4),  18  x"^  -  Ibxy  +  Sy^  =  0.  (5) 

Factor,  (y  -  3  x)  (y-2x)  =  0.  (6) 


316  ELEMENTS   OF  ALGEBRA 

System,  (6)  and  (1),  which  is  equivalent  to  (a),  is  equivalent  also 
to  the  two  systems  (&). 

x2  +  ce?/  +  2  y2  =  44,  I  x2  +  a;?/  +  2  2/2  =  44,  | 

y-^x  =  0.  J  y-'2x  =  0.  J  ^^^ 

The  solutions  of  systems  (&)  are  y/2,  Zy/2;  —y/2,  —3^2;  2,  4; 
and  —  2,  —  4  ;  which  are  therefore  all  the  solutions  of  (a). 

Ex.  2.    Solve  the  system         ?/2  _  2  a:2  =  4  x,  (1)  ] 

3  ?/2  +  xz/  -  2  a;2  =  16  x.  (2)  J  ^^^ 

The  terms  below  the  2d  degree,  4  x  and  16  x,  are  similar. 
We  proceed  to  eliminate  the  term  in  x. 

Multiply  (1)  by  4,  4  ?/2  _  8  a;2  =  16  x.  (3) 

Subtract  (2)  from  (3),      y^  -  xy  -  Qx'^  =  0. 

Factor,  (y  +  2  x)  (y  -  3  x)  =  0.  (4) 

System,  (4)  and  (1),  which  is  equivalent  to  system  (a),  is  equiva- 
lent also  to  the  two  systems  (6)  and  (c). 

?/2  _  2  a:2  =  4  X,  ]  ?/2  -  2  ^2  =  4  x. 


]  2/'-2x2  =  4x,  1 


?/  +  2x  =  0.         '   ^"^  ..      o^      A  1   (<^) 

The  two  solutions  of  system  (h)  are  0,  0  and  2,  —  4 ;  those  of  (c) 
are  0,  0  and  4/7,  12/7  ;  which  are  therefore  the  four  solutions  of  (a). 

Observe  that  by  eliminating  the  term  below  the  second 
degree  in  each  of  the  systems  above,  we  obtained  a  homo- 
geneous equation  in  x  and  y,  which  we  resolved  into  two 
equivalent  equations. 

Instead  of  eliminating  the  term  below  the  second  degree, 
it  is  sometimes  better  to  eliminate  one  of  the  terms  of  the 
second  degree. 

Ex.  3.    Solve  the  system    9  x2  -  8  y2  ^  28,  (1)  ] 

7x2 +  3  2/2  =  31.  (2)  J  '^^'^ 

Multiplying  (1)  by  3  and  (2)  by  8,  and  adding,  we  eliminate  y"^ 

and  obtain 

83  x2  =  332,  or  X  =  ±  2.  (3) 

When  X  =  2,  from  (2)  we  obtain       y  =  ±\. 

When  X  =  —  2,  from  (2)  we  obtain  y  =±l. 


SYSTEMS   OF  QUADRATIC  EQUATIONS  317 

Hence  the  four  solutions  of  (a)  are  2,  1 ;  2,  —  1 ;  —  2,  1 ;  —2,-1. 

Ex.  4.   Solve  the  system      xy  +  x  =  25,  (1)  | 

2xy-Zy  =  2S.  (2)  J 
Eliminating  the  product  xy  we  obtain 

2  X  +  3  2/  =  22.  (3) 

Solving  system,  (1)  and  (3),  which  is  equivalent  to  system  (a),  we 
obtain  the  two  solutions  5,  4  ;  16/2,  7/3. 

Ex.  5.    Solve  the  system    x^  —  Sxy  =  10,  (1)  ] 


4y2-xy  =  -l.  (2) 


|(«: 


Sometimes  by  adding  or  subtracting  the  given  equations  we  obtain 
an  equation  which  can  be  resolved  into  equivalent  equations. 

Add  (1)  and  (2) ,     x:^  -  4xy  +  iy^  =  9, 

or  x-2y=±S.  (3) 

System,  (1)  and  (3),  which  is  equivalent  to  system  (a),  is  equiva- 
lent to  the  two  systems  (6)  and  (c) . 


x^-Sxy  =  lO,]  x^-Sxy  =  lO, 

x-2y  =  S.        >^'^  -      "-         '^   ^^'^ 


^-Sxy  =  lO,    I 
x-2y=-3.  J 


Exercise  115. 
Solve  each  of  the  following  systems  of  equations : 


1. 

x^  +  xy=12, 
x!/-if  =  2.   J 

6. 

ar^  +  52/2  =  84, 

3a^  +  17a'?/  +  84  =  2/^ 

2. 

x^-{-xy  =  24, 
2f-{-3xy  =  S2. 

7. 

ar-7xy-9y-  =  9, 

x'^5xy-\-lly'  =  5 

3. 

ic2  +  3a«/  =  7, 
f  +  xy  =  6.    . 

8. 

x(x-^y)  =  AO, 

\ 

y(x-y)=:6.   J 

4. 

2,x?-hf  =  2%\ 
Sxy-4:f  =  S.  J 

9. 

x^  +  xy  +  f  =  7, 

6x'-2xy-\-y'  =  i5.. 

5. 

x^-Sxy-^2f  =  3 

y 

10. 

x'-^3xy  =  2S,\ 

2x'  +  y'  =  6. 

xy-\-4:y'  =  S.    > 

818 


ELEMENTS   OF  ALGEBRA 


In  example  10,  add  the  two  equations. 


=  40,1 
=  9.    J 


11.  a^  +  3aj?/  =  40, 
4  ?/2  +  0^2/ 

12.  a^  +  3fl7?/  =  54, 
xy  -\-4:y^  =  115. 


13.  x^-\-xy-\-A4:  =  2y^,) 
xy  +  3  /  =  80.         J 

14.  3xy-{-x'^  =  10, 
5xy  —  2  x'^  =  2.. 


In  example  14,  eliminate  x^  or  the  product  xy. 


=  301.  J 


15.  Ax'-Sy^^-ll, 
11  a^  +  5  2/' 

16.  2x^-\-y^  =  9, 
5aP-{-(yy'  =  26.} 

17.  20  a^- 16/ =  179, 
5a;2-336/  =  24. 

18.  2a^-2xy-3y^  =  lS, 
3x'-2y'  =  19. 

19.  i»2  4-3a;  — 2?/  =  4 


2x'-5x-{-3y 


u\ 


20.  (a;  +  l)(2/  +  l)  =  10, 
a?2/  =  3. 

21.  4:x'-3xy  =  10, 
y^-xy  =  6. 

22.  x^  —  2xy  =  3y, 
2x'-9y''  =  9y. 

23.  2x^-xy-^y^  =  2y; 
2  a^  -]-  4:  xy  =  5  y. 

24.  a.'3  +  l  =  9?/,| 
a^  4-  a;  =  6  2/.  J 


308.  Systems  of  symmetrical  equations.  A  symmetrical 
equation  is  one  which  is  not  changed  by  interchanging  its 
unknowns. 

E.g.,  x-\-  y  =  12,  xy  =  35,  x^  +  y^  _  74^  x^±2  xy  +  y^  =  16  arc 
symmetrical  equations.  The  equations  x  —  y  =  2,  x^  —  y^  =  4, 
x^  —  y^  =  IQ  are  symmetrical  except  for  sign. 

The  methods  given  below  for  solving  systems  of  sym- 
metrical equations  can  usually  be  employed  when  the  equa- 
tions are  symmetrical  except  for  sign. 


(a) 


Ex.  1.    Solve  the  system, 

X^  +  y2  =  74, 

(1) 

xy  =  35, 

(2) 

Multiply  (2)  by  2, 

2xy  =  70. 

(3) 

SYSTEMS  OF  QUADRATIC  EQUATIONS 


319 


Add  (3)  to  (1),  x^-\-2xy  +  y^  =  144, 

x  +  y  =  ±l2.  (4) 

Subtract  (3)  from  (1),  x-y  =  ±2.  (6) 

By  §  301,  system  (6)  is  equivalent  to  the  four  systems  (c) . 


a;  +  y  =  12, 
x-y  =  2. 


X  4-  y  =  12, 
x-y=-2. 


+  i/=-12,  I     x  +  y  =  -l2, 
-y  =  2.       J      x-y  =  -2. 


The  solutions  of  systems  (c)  are  7,  5  ;  5,  7  ;  —  5, 

Ex.  2.  Solve  the  system,  x^  -  xy  -\- ij^  =  49, 

x  +  y=lS. 


or 


Square  (2), 

Subtract  (1)  from  (3), 

Subtract  (4)  from  (1), 


x2  +  2  xy  +  1/2  =  169. 
3  xy  =  120, 
xy  =  40. 
x-y=±3. 

System,  (2)  and  (5),  is  equivalent  to  the  two  systems  (6). 

a;  +  y=13,    1 


(1) 
(2) 
(3) 


(&) 

(c) 
5. 

) 


|(« 


(4) 
(5) 


a:  +  y  =  13,  1 
x-y  =  3.    J 


a;  _  y  =  _  3.   J 


(6) 


The  solutions  of  systems  (6)  are  8,  6,  and  5,  8. 

The  four  solutions  of  system,  (1)  and  (5),  must  include  the  two 
solutions  of  (a),  since  no  solution  was  lost  by  squaring  (2). 

Hence  the  two  solutions  of  (a)  must  satisfy  (2)  and  also  (5). 

Therefore  the  solutions  of  systems  (b)  are  the  two  solutions  of  (a). 

Observe  that  each  of  tlie  above  systems  was  solved  by  first  finding 
the  values  of  x  +y  and  x  —y. 


Ex.  3.     Solve  the  system    x*  +  y<  =  82, 
x-y  =  2. 

Let  x  —  v  +  w^ 

and  y  =  v  —  w. 

From  (2),  (3),  and  (4),  w=\. 

From  (1),  (3),  (4),  and  (6), 

(v+l)*+  (tj-l)*  =  82, 
or  (r2  +  i0)(t;2_  4)3,0. 

.•.v  =  ±2,  or±^^ 


10. 


(1) 

(2) 

(3) 
(4) 
(5) 


(6) 


1" 


(ft) 


320  ELEMENTS   OF  ALGEBRA 

From  (3),  (5),  and  (6),  x  =  S,  -1,  \  ±  V^^TO. 


From  (4),  (5),  and  (6),  ^  =  i,  _  3,  _  i  ^  V^TTo. 


(c) 


System  (a)  with  (3)  and  (4)  forms  a  system  equivalent  to  (6), 
which  is  equivalent  to  (c)  with  (5)  and  (6). 

Hence  the  four  solutions  of  (a)  are  given  in  (c). 


Exercise  116. 


Solve  each  of  the  following  systems  of  equations  by  first 
finding  the  values  oi  x  -]-  y  and  x  —  y : 

1.    a^2_^/  =  89,  1  5.    x'^l+y'  =  Zxy, 


xy  =  4.0.         J  Ss(f  —  xy-{-3y^  =  13. 


2.    x'-\-y'  =  170y 
xy  =  13. 


6.    ^ 


^  -  ^2/  +  r  =  "6, 1 

a;  +  .?/  =  14.  J 


3.  x'  +  y^=Qo,\  7.    x'  +  xy  +  f^^QlA 
iC2/  =  28.         J  .'c  +  ?/  =  9.  J 

4.  a^  +  ic2/  +  2/^  =  67,  1  8.    x"  -  4.xy +  y~  =  b2,\ 
a^-^^  +  2/'  =  39.J  _i^(ar_2/)  =  l.  J 

9.  Solve  the  systems  in  examples  1,  2,  4,  5,  8,  12,  and  14 
in  exercise  113,  by  first  finding  the  values  oi  x -\- y  and 
x-y. 

Solve  each  of  the  following  systems  of  equations : 

10.  x-.y  =  3,  1  13.    x^  +  y^  =  212A 
a;2  _  3  ic2/  +  /  =  -  19.  J  x-y  =  2.        J 

11.  x^  —  xy-\-y'-=12A  14:.    x  —  y  =  2,        1 
x-{-y  =  lA.             J  x'  —  jf  =  242.  J 

12.  a; +  2/ =  4,      1  15.    aj^  4- ^Z'' =  706, 


J 


a;4  _j_  ^4  ^  82.  j  a;  +  2/  =  8. 

16.    Solve  system  (2)  in  §  263,  and  observe  that  x  and  y 
are  rational  only  when  a^  —  6  is  a  perfect  square. 


SYSTEMS   OF  QUADRATIC  EQUATIONS  321 

309.  Division.  If  the  members  Qf  one  equation  (1)  are 
divided  hy  the  corresponding  memhers  of  another  equation 
(2),  and  the  derived  equation  (3)  is  integral  in  the  unknowns; 
then  the  system  (a)  is  equivalent  to  the  two  systems  (6)  and  (c). 

AB  =  A'B',   (1)1  A  =  A\   (3)1  5  =  0,    (5)1 

B  =  B'.        (2)r  ^       B  =  B'.    (4)  J  ^  ^     B'  =  0.    (6)  J  ^  ^ 

Observe  that  (3)  is  the  derived  equation,  that  (4)  is  the 
same  as  (2),  and  that  (5)  and  (6)  are  formed  by  equating 
to  0  the  members  of  (2). 


Proof.     Substituting  B  for  B'  in  (1)  we  obtain  the  sys- 
tem (d). 

B(A-A')  =  0, 

B  =  B'. 


)W 


By  §  202,  system  (d)  is  equivalent  to  system  (a). 
By  §  304,  system  (cl)  is  equivalent  to  the  two  systems  (6) 
and  (e). 

■8  =  0,  \ 

By  substitution  (§  202),  system  (e)  is  equivalent  to  (c). 
Hence  (a)  is  equivalent  to  the  two  systems  ip)  and  (c). 


E.g.^  dividing  (1')  by  (2')  we  obtain  the  integral  equation  (3'); 
y^  =  x{x^y),  (1') 

y^  =  X  +  y.  (2 

Hence  system  (a')  is  equivalent  to  the  two  systems  (6')  and  (c') 


;;}- 


V  =  -,  (30  I  .^  =  0,        (50  1 

y^  =  ic  +  y.        (40J  j;  +  y  =  0.        (60  J 

Whenever  the  equation  B  =  0  or  JB'  =  0  is  impossible,  sys- 
tem (c)  will  be  impossible,  and  system  (a)  will  be  equivalent 
to  system  (6), 


(a) 


5)  J 


822  ELEMENTS   OF  ALGEBRA 

Ex.  1.   Solve  the  system        a;^  -y^  =  21,  ^^^  \ 

X  -  y  =  3,  (2)  J 

Dividing  (1)  by  (2)  we  obtain  the  integral  equation  (3);  hence, 
as  B'  =0,  or  3  =  0,  is  impossible,  system  (a)  is  equivalent  to  (6). 

x2  +  x^  +  if  =  9,  (3) 

x-y  =  S.  (4) 

Ex.  2.    Solve  the  system  x*  +  x^  -\.  i/  =  lii7\,  (1) 

x2  _  a;i/  +  1/^  =  03.  (2) 

■    Divide  (1)  by  (2),  x^  +  xy  +  y^  =  117.  (3) 

Add  (2)  and  (3),  x'^  +  y-  =  CO,  (4)  ] 

Subtract  (2)  from  (3),  •2xy  =  54.  (5 

Since  03  =  0  is  impossible,  by  division  (§  309)  and  addition  (§  204) 
system  (&)  is  equivalent  to  (a). 

Ex.  3.    Solve  the  system  x-y  +  xy-  =  30,  (1) 

x-\-y  =  5.  (2) 

Divide  (1)  by  (2),  xy  =  0.  (3) 

Equations  (2)  and  (3)  form  a  system  equivalent  to  (a). 


Exercise  117. 
Solve  each  of  the  following  systems  : 
1.   a^  +  2/'  =  3473,  1  6.    x' +  x'y' +  y' =  2^3 

x-y  =  4..        i  a^-xy-^y'  =  S7 


(a) 


a;  +  ?/  =  23.        J  x'-xy+y'  =  9.        J 

2.  a.-^  -  2/' =  218, 1  7.    x*  +  x^ -h  y' =  91, ) 
x-y  =  2.        J  aj2+ a.'^  +  2/' =  13.    J 

3.  x'-7f=^9SS,\  8.    a;^-fa^/  + ^'  =  2923,1 


^3  -  7/3  ^  2197, 1  9.   x^-\-a^y^-{-y'  =  7371, 

x-y  =  13.        J  a.-2-j'?/  +  2/2  =  63. 

a..4  +  ^2^2  _^  2/^  ^  2128, 1  10.    x'-f=56,              1 

a;2  4_  0^2/ +  2/' =  "6.        J  x" -\- xy -{- y' =  2S.    J 


SYSTEMS   OF  QUADRATIC  EQUATIONS 


823 


11.   a^  +  f  =  126,         I  12.    x-^y-Vxy  =  7,     1 

ay'-xy  +  f  =  21.S  ^^fj^xy=^  133.  J 

In  the  next  four  systems  apply  §  304  first. 

13.  a; +  2/ =  5,  1  15.    a;  +  2/  =  l, 
4  icy  =  12  -  x'y\  J  x'y^  +  13  rt^  +  12  =  0.  J 

14.  a?y-\-xf  =  imA  16.    Sa^  -  5/  =  a; +  ?/,  1 
a2/=400.           J  3a;2_32^2^3._2,  J 

In  the  next  four  systems  let  the  unknowns  be  the  reciprocals  of  x 
and  y,  and  let  v  =  \/x  and  w  =  \/y. 


17.  2/.^  +  1/?/ =  1, 

^  +  ^  +  i=5. 
y?     xy     y"^ 

18.  l/a;  +  l/7/  =  2, 
1/a^  4- 1//  =  20.  J 

19.  3/ar^-l//  =  l, 

^-1  +  1  =  3. 
a?     xy     f 


20. 


21. 


l/x2-l/(4y^) 

i_l  +  J_ 

ar*     a^     42/^ 

x_-j-j  x—ji 
x-y  x+y 
x'  +  f  =  20. 


3, 


=  9. 


22.    a^A  +  /A=9/2, 
a;  +  2/  =  3. 


310.  It  should  be  observed  that  the  methods  given  in  this 
chapter  are  applicable  only  to  special  systems  of  quadratic 
and  higher  equations,  and  do  not  enable  us  to  solve  a  sys- 
tem of  any  two  quadratic  equations ;  for  the  equation  de- 
rived by  eliminating  one  unknown  will,  in  general,  be  above 
the  second  degree  in  the  other  unknown,  and  we  have  not 
yet  learned  how  to  solve  an  equation  of  a  higher  degree 
than  the  second,  except  in  very  special  cases. 

E.g.^  consider  the  system 

a;2  +  x  +  y  =  3,  x2  +  y2  =  5.  („) 

Solving  the  first  equation  for  y  and  substituting  its  value  in  the 

second,  we  have 

a;-2  +  (3  -  y  -  x)2  =  5, 

or  x4  +  2a;8-4x-^ -6a;  +  4  =  0.  (1) 

Equation  (1),  which  is  of  the  fourth  degree,  cannot  be  solved  by 
any  methods  which  have  been  given  in  the  previous  chapters. 


324  ELEMENTS  OF  ALGEBRA 

Exercise  118. 

1.  The  difference  of  two  numbers  is  7,  and  the  sum  of 
their  squares  is  169.     Find  the  numbers. 

2.  The  sum  of  the  squares  of  two  numbers  is  130,  and 
the  difference  of  their  squares  is  32.     Find  the  numbers. 

3.  The  sum  of  two  numbers  is  39,  and  the  sum  of  their 
cubes  is  17,199.     Find  the  numbers. 

4.  A  person  bought  some  fine  sheep  for  $  360,  and  found 
that  if  he  had  bought  6  more  for  the  same  money,  he  would 
have  paid  $  5  less  for  each.  How  many  did  he  buy,  and 
what  was  the  i)rice  of  each  ? 

5.  If  the  length  and  breadth  of  a  rectangle  were  each 
increased  by  1  yard,  the  area  would  be  48  square  yards ;  if 
they  were  each  diminished  by  1  yard,  the  area  would  be  24 
square  yards.     Find  the  length  and  breadth. 

6.  The  numerator  and  denominator  of  one  fraction  are 
each  greater  by  1  than  those  of  another,  and  the  sum  of  the 
two  is  1  j^ ;  if  the  numerators  were  interchanged,  the  sum 
of  the  fractions  would  be  1^.     Find  the  fractions. 

7.  For  a  journey  of  108  miles,  6  hours  less  would  have 
sufficed,  had  the  traveller  gone  3  miles  an  hour  faster.  At 
what  rate  did  he  travel  ? 

8.  The  hypotenuse  of  a  right-angled  triangle  is  20  feet, 
and  its  area  is  96  square  feet.  Find  the  length  of  the  other 
two  sides. 

9.  A  number  is  divided  into  two  parts  such  that  the 
sum  of  the  first  and  the  square  of  the  second  is  twice  the 
sum  of  the  second  and  the  square  of  the  first ;  and  the  sum 
of  the  number  and  the  first  part  is  4  more  than  twice  the 
second.     Find  the  number. 

10.   The  small  wheel  of  a  bicycle  makes  135  revolutions 
more  than  the  large  wheel  in  a  distance  of  260  yards;  if 


SYSTEMS   OF  QUADRATIC  EQUATIONS  325 

the  circumference  of  each  were  one  foot  more,  the  small 
wheel  would  make  27  revolutions  more  than  the  large  wheel 
in  a  distance  of  70  yards.  Find  the  circumference  of  each 
wheel. 

11.  A  man  bought  6  ducks  and  2  turkeys  for  $  15.  For 
$  14  he  could  buy  4  more  ducks  than  he  could  turkeys  for 
$  9.     Find  the  price  of  each. 

12.  The  sum  of  the  cubes  of  two  numbers  is  407,  and  the 
sum  of  their  squares  exceeds  their  product  by  37.  Find 
the  numbers. 

13.  A  rectangular  field  contains  160  square  rods.  If  its 
length  be  increased  by  4  rods,  and  its  breadth  by  3  rods,  its 
area  will  be  increased  by  100  square  rods.  Find  the  length 
and  breadth  of  the  field. 

14.  A  man  rows  down  stream  12  miles  in  4  hours'  less 
time  than  it  takes  him  to  return.  Should  he  row  at  twice 
his  ordinary  rate,  his  rate  down  stream  would  be  10  miles 
an  hour.  Find  his  rate  in  still  water,  and  the  rate  of  the 
stream. 

15.  The  sum  of  two  numbers  is  7,  and  the  sum  of  their 
fourth  powers  is  641.     Find  the  numbers. 

16.  A  gentleman  left  $  210  to  3  servants  to  be  divided  in 
continued  proportion,  so  that  the  first  should  have  $  90 
more  than  the  last.     Find  the  legacy  of  each. 

17.  From  a  sheet  of  paper  14  inches  long,  a  border  of 
uniform  width  is  cut  away 'all  round  it,  and  the  area  is 
thereby  reduced  |;  but  had  the  sheet  been  3  inches  nar- 
rower, and  a  border  of  the  same  width  had  been  cut  away, 
the  area  would  have  been  reduced  ^.  Find  the  breadth  of 
the  paper,  and  the  width  of  the  border  cut  away. 

18.  A  and  B  set  out  from  the  same  place,  and  travel  in 
the  same  direction  at  uniform  rates.     B  starts  5  hours  after 


326  ELEMENTS   OF  ALGEBRA 

A,  and  overtakes  him  after  travelling  100  miles.  Had  their 
rates  of  travelling  been  a  mile  per  hour  less,  B  would  have 
overtaken  A  after  travelling  60  miles.     Find  their  rates. 

19.  A  man  has  to  travel  a  certain  distance,  and,  when  he 
has  travelled  40  miles,  he  increases  his  speed  2  miles  per 
hour.  If  he  had  travelled  with  his  increased  speed  during 
the  whole  journey,  he  would  have  arrived  40  minutes 
earlier;  but  if  he  had  continued  at  his  original  speed,  he 
would  have  arrived  20  minutes  later.  Find  the  whole  dis- 
tance he  had  to  travel,  and  his  original  speed. 

20.  A  cubical  tank  contains  512  cubic  feet  of  water.  It 
is  required  to  enlarge  the  tank,  the  depth  remaining  the 
same,  so  that  it  shall  contain  7  times  as  much  water  as 
before,  subject  to  the  condition  that  the  length  added  to 
one  side  of  the  base  shall  be  4  times  that  added  to  the 
other.     Find  the  sides  of  the  new  rectangular  base. 


CHAPTER  XXIV 
INEQUALITIES 

311.  An  inequality  is  the  statement  that  one  number  is 
greater  or  less  than  another,  as  6  >  4,  —  3  <  —  2.  See 
§§  7  and  31. 

312.  When  a  and  h  are  real,  in  §  31  we  agreed  to  say 
that : 

a  >  6,  when  a  —  6  is  positive; 

and  a<b,  when  a  —  b  is  negative. 

The  statement  'a  —  b  is  positive '  is  expressed  in  symbols 
by  (I  —  ft  >  0 ;  and  'a  —  b  is  negative '  by  a  —  6 <  0. 

In  this  chapter  we  shall  not  consider  imaginary  or  complex 
numbers. 

313.  Two  inequalities  are  said  to  be  like  or  unlike  in 
species  according  as  they  do  or  do  not  have  the  same  sign  of 
inequality. 

E.g.y  the  inequalities  8>4  and  a>b  are  like  in  species;  while 
2  <  3  and  «  >  6  are  unlike  in  species. 

If  a  >  6  ;  then,  conversely,  b<a. 

The  inequality  a > 6  and  its  converse  b<a  are  unlike. 

314.  Principles  of  inequalities. 

(i)  If  one  number  >  a  second,  and  this  second  number  >  a 
third,  then  the  first  number  >  the  third  number. 

That  is,  if  a  >  6  and  6  >  c,  then  a'^c. 

(ii)  If  the  same  number  is  added  to  both  members  or  sub- 
tracted from  both  members  of  an  inequality,  the  derived  in- 
equality will  be  like  the  given  one. 

That  is.  if  a >  ft,  then  a  ±7)i>b  ±  m. 

327 


328  ELEMENTS  OF  ALGEBRA 

(iii)  If  the  corresjyonding  members  of  two  or  more  like 
inequalities  are  added,  the  derived  inequality  will  he  like  the 
given  ones. 

That  is,  if  a  > 6  and  c > <?,  then  a -f  c> 6  +  d. 

(iv)  If  both  members  of  an  inequality  are  multiplied,  or 
divided,  by  the  same  positive  number,  the  derived  inequality 
will  be  like  the  given  one. 

That  is,  if  a>b,  then  a  (+n)  >  b  (^w),  or  a-r-'^n^b-h  '^n. 

(v)  If  both  members  of  an  inequality  are  multiplied,  or 
divided,  by  the  same  negative  number,  the  derived  inequality 
will  be  unlike  the  given  one. 

That  is,  if  a > 6,  then  a (~n) < b (~n),  or  a-i-~n<b-r- ~n. 

(vi)  If  all  the  members  of  two  or  more  like  inequalities  are 
positive,  and  if  the  corresponding  members  are  multiplied 
together,  the  derived  inequality  will  be  like  the  given  ones. 

That  is,  if  +%>+&!,   ^a2>+bz,  ••♦, 
then  +ai  • +a2--->+6i  • +62"-. 

(vii)  If  both  members  of  an  inequality  are  positive,  and  they 
are  raised  to  the  same  positive  integral  power,  the  derived 
inequality  ivill  be  like  the  given  one. 

That  is,  if  ^a  >+6,  then  ("^a)**  >  (+6)%  where  w  is  a  positive 
integer. 

(viii)  If  the  same  principal  roots  of  both  members  of  an 
inequality  are  taken,  the  derived  inequality  will  be  like  the 
given  one. 

That  is,  if  a  >  6,  then  ^a  >-^&. 

Proof  of  {}).     (a  —  6)  +  (6  —  c)=a  —  c. 
Hence,  if  a  —  6  >  0  and  6  —  c>  0,  then  a  —  c  >  0 ; 
that  is,  if  a>h  and  & > c,  then  a>c. 

Proof  of  (it),     a  —  b  =  (a  ±  m)  —  (b  ±  m). 
Hence,  if  a  >  6,  then  a  ±m>b  ±  m. 
The  proof  of  the  other  principles  is  left  as  an  exercise  for  the  pupil. 


INEQ  UA  LITIS  S  329 

316.   The  following  principle  is  often  useful  in  proving 
inequalities : 

Jfa  and  b  are  unequal  and  realy  a-  +  6-  >  2  ab. 

Proof.  (a-by>0, 

or  a^-2ab-\-b'>0.  (1) 

Adding  2  a6  to  each  member,  by  (ii)  of  §  314  we  obtain 
a^  +  b'>2ab,  when  a  ^  b.  (2) 

Observe  that  a^  and  b'  are  both  positive. 

Ex.  1.     Prove  (x  +  y)/2>y/xy,  if  a; > 0,  y  > 0,  and  x  #  y. 
If  in  (2)  we  put  x  for  a*  and  y  for  6*,  we  obtain 

X  -\-y>2 y/xy. 
Hence,  by  (iv),  (x  +  y)/2  >\/xy,  where  x  >  0,  y  >  0,  and  x  :^  y. 

Ex.  2.     a«  +  6'  >  a^b  +  ab\  if  «  +  6  >  0  and  a  ^  6. 

From  (1),  a^  -ab  +  6^  >  ah.  by  (ii) 

Multiply  by  a  +  6,      a^  ■¥b^>  a'^b  +  aft-^.  by  (iv) 

Ex.  3.     The  sum  of  any  positive  number,  except  1,  and  its  recipro- 
cal is  greater  than  2. 

Let  the  number  be  n ;  then  in  (2),  putting  n  for  a^  and  1/n  for 

62  we  obtain 

n  +  l/«>2. 

316.   The  following  examples  illustrate  some  of  the  uses 
of  the  principles  of  inequalities : 

Ex.  1.   For  what  values  of  x  is  (6x  -  7)/3  >(2  -  3  x)/5  ?         (1) 

Multiply  by  15,  25  x  -  35  >  6  -  9  x.  by  (iv) 

Transpose,  34x>41.  by  (ii) 

Divide  by  34,  »>  41/34.  by  (iv) 

Hence  (1)  is  satisfied  for  any  value  of  x  gi-eater  than  41/34. 

Ex.  2.    For  what  values  of  x  is  x^  -  4  x  +  3  >  -  1  ?  (1) 

Addl,  x2-4x  +  4>0,  or  (x-2)2>0.  by  (ii) 

Hence  (1)  is  satisfied  when  (x  —  2)2  >0,  i.e..,  when  x  has  any  real 
value  except  2. 


330  ELEMENTS   OF  ALGEBRA 

Ex.  3.   Find  what  values  of  x  satisfy  the  inequalities 

4iK-6<2x  +  4,  (1)  \ 

and                                       2  X  +  4  >  16  -  2  ic.  (2)  J 

From  (1),                        2x<10,  or  x<5.  by  (ii),  (iv) 

From  (2),  4ic>12,  or  ic>3.  by  (ii),  (iv) 
Hence  (1)  and  (2)  are  satisfied  by  any  value  of  x  between  3  and  5. 
Ex.  4.   Find  what  values  of  x  satisfy  the  inequality 

x^-lx<S.  (1) 

Subtract  8,   x2  -  7  ic  -  8  <  0,  or  (x  -  8) (x  +  1)< 0.  by  (ii) 

The  product  (x  -  8)  (x  +  1)  will  be  negative,  when,  and  only  when, 
one  factor  is  positive  and  the  other  negative. 

One  of  these  factors  will  be  positive  and  the  other  negative  when  x 
has  any  value  between  —1  and  8,  and  only  then. 

Hence  (1)  is  satisfied  by  any  value  of  x  between  —  1  and  8. 

Ex.  5.   Find  what  values  of  x  and  y  satisfy  the  inequality 

3x  +  2?/>5,  (1)| 

and  the  equation                        6  x  -\-  7  y  =  12.  (2)  J 

Multiply  (1)  by  5,             15  x  +  10  2/  >  25.  (3) 

Multiply  (2)  by  3,             15  x  +  21  y  =  36.  (4) 

Subtract  (4)  from  (3),              -  11  y  >-  11,  or  ?/  <  1.  by  (v) 

Multiply  (1)  by  7,               1  x  +  14  y  >  35.  (5) 

Multiply  (2)  by  2,  10  x+Uy  =  24.  (6) 
Subtract  (6)  from  (5),                 11  x  >  11,  or  x  >  1. 

Hence  any  solution  of  equation  (2)  in  which  x  >  1  and  y  <  1  will 

satisfy  both  (1)  and  (2). 

Exercise  119. 
If  the  letters  denote  unequal  positive  numbers,  prove : 

1.  a^-\-b'^-\-c^>ah  +  ac  +  bc.  (1) 
Use  the  relation  a^  4.  52  ->  2  ab. 

2.  a^  +  b''>d'b-{-ab\  (2) 


INEQUALITIES  331 

3.  ^>^;  %  +  -A+--  <3) 

4.  a^j^ly'  +  (?>  (a'b  +  ab^  +  a-c  +  ac"  +  b^c  +  5c2)/2. 

5.  am  +  67i+cr<l,  if  a^-{-b^-]-(^=zl,  and  m^+7i^+)'^=l. 

Find  the  limits  between  which  the  values  of  x  must  lie 
to  satisfy  each  of  the  following  inequalities : 

6.  6  a;  >  fa;  4- 18.  10.  a;^  _^  a;  >  12. 

7.  ^a;-|a;>|a;-3.  11.  (a;  +  2)/(a;  -  3)  >  0. 

8.  -2(a;  +  7)>-16.  12.  (a;  -  7) /(a;  +  4)  <  0. 

9.  x^  —  ox>  —  4:.  13.  3(a;+7)/5>5(a;-3)/7. 

14.  If  5  a;  —  6  <  3  a;  +  8  and  2a;  +  l<3a;  —  3,  show  that 
the  values  of  x  lie  between  4  and  7. 

15.  If  3  a;  —  2  >  I  a;  —  I  and  J  —  fa;<8  —  2  a;,  show  that 
the  values  of  x  lie  between  12/25  and  82/9. 

Find  what  values  of  x  and  y  will  satisfy  each  of  the 
following  systems : 

16.  2x-\-3y=4:,]      17.   Sx-y=Q,)      18.    4:X-2y=e, 

x—y>2.)  2x+y>4:.)  2x—Sy>5. 

19.  Show  that  (1)  in  example  1  holds,  if  a,  6,  and  c  are 
real  and  either  a  ^  b,  or  a  ^  c,  or  b  ^  c. 

20.  Show  that  (2)  and  (3)  in  examples  2  and  3  hold,  if  a 
and  b  are  real  and  unequal  and  a-\-b>0. 


CHAPTER   XXV 
RATIO  AND  PROPORTION 

317.  The  ratio  of  one  number  to  another  is  the  quotient 
of  the  first  divided  by  the  second. 

The  dividend  is  called  the  first  term,  or  the  antecedent,  of 
the  ratio ;  and  the  divisor,  the  second  term,  or  consequent. 

The  ratio  of  a  to  ft  is  written  -,  a/b,  a  -^  b,  or  a:  b,  each  of  which 

b 
forms  can  be  read  '  a  is  to  6  '  or  '  a  by  &.' 

The  ratio  of  8  to  2  is  8/2,  or  4  ;  the  ratio  of  7  to  5  is  7/5. 

It  is  clear  that  a  ratio  is  arithmetically  greater  than,  equal 
to,  or  less  than  1,  according  as  its  first  term  is  arithmetically 
greater  than,  equal  to,  or  less  than,  the  second. 

318.  Since  a  ratio  is  a  fraction,  all  the  properties  of  frac- 
tions belong  to  ratios  in  whatever  form  the  ratios  are  written. 

Thus  a:b  =  am:bm,  or  a/b  =  am/(bm) ;  §172 

and         a  :  b=(a  -^  m)  :  (b  -^  m),  or  a/b={a  -^  m)/(b  -=-  m).     §  173 

Two  ratios  can  be  compared  by  reducing  them  as  fractions 
to  a  common  denominator. 

Ex.  1.    Which  is  the  greater,  3:11  or  5  :  19  ? 

3:11=  3/11  =  57/209,  and  5  :  19  =  5/19  =  55/209 ; 
hence  the  ratio  3  :  11  >  the  ratio  5  :  9. 

Ex.  2.    (a  :  bf  =  a^  :  b^  ;  V^76  =  V«  -  V&-  §§  186,  225. 

319.  By  §  91,  (a  :  &)  (c  :  d)  (e  :/)  =  ace  :  hdf. 

The  ratio  ace :  hdf  is  said  to  be  compounded  of  the  ratios 
a:b,  c:  d,  and  e  :  /. 


RATIO  AND   PROPORTION 

320.  The  inverse  of  a  ratio  is  its  reciprocal. 

Hence  the  inverse  of  the  ratio  a :  6  is  the  ratio  b :  a 
(§  182). 

321.  By  §  183,  a/b  :c/d  =  ad:  be. 

Hence  the  ratio  of  any  two  fractions  can  be  expressed  by 
the  ratio  of  two  integers. 

322.  Two  numbers  are  said  to  be  commensurdble  or  incoiii- 
mensurable  with  each  other  according  as  their  ratio  can  or 
cannot  be  expressed  by  the  ratio  of  two  integers. 

E.g.,  y/2  and  5  are  incommensurable  with  each  other,  so  also  are 
^3  and  ^5.  The  incommensurable  numbers  3^2  and  7v^2  are  com- 
mensurable with  each  other;  for  their  ratio  is  3/7.     Compare  §  224. 

323.  Ratio  of  concrete  quantities.  If  A  and  B  are  two  con- 
crete quantities  of  the  same  kind,  whose  numerical  measures 
in  terms  of  the  same  unit  are  the  numbers  a  and  6,  then  the 
ratio  of  ^  to  J5  is  defined  to  be  the  ratio  of  a  to  b. 


Exercise  120. 
Find  the  simplest  expressions  for  the  following  ratios : 

1.  6  a  to  12  a'*.  4.   a/x  to  c/y. 

2.  3  a^x/5  to  6  aa^/J.  5.    a/{x  -  2)  to  S/(x  -  2y. 

3.  1/a  to  1/6.  6.   9/(a-by  to  6/(a-b). 

7.  Write  as  a  ratio  (2x:Syy;  (2a:  bf ;  («  :  c)« ;  i^oTb. 

Find  the  ratio  compounded  of : 

8.  The  ratio  25  :  8  and  the  square  of  the  ratio  4  :  3. 

9.  The  ratio  32  :  27  and  the  cube  of  the  ratio  3  :  2. 

10.  The  ratio  6 :  7  and  the  square  root  of  the  ratio  25 :  36. 

11.  Arrange  the  ratios  5  :  6,  7  :  8,  41 :  48,  and  31 :  36  in 
descending  order  of  jnagnitude 


334  ELEMENTS   OF  ALGEBRA 

12.  For  what  value  of  x  will  the  ratio  15  + a;:  17  + a? 
be  equal  to  1/2  ? 

13.  What  number  must  be  added  to  each  of  the  terms  of 
the  ratio  3  :  4  to  make  it  equal  to  the  ratio  25  :  32  ? 

Let  X  =  the  number  to  be  added  ;  then 

(3  +  a;)/(4  +  a;)=25/32. 

14.  Find  two  numbers  in  the  ratio  of  5  to  6,  whose  sum 
is  121. 

15.  Which  is  the  greater  ratio,   5:7  or  5  +  2:7  +  2? 

16.  Which  is  the  greater  ratio,   7:5  or  7  +  2:5  +  2? 

PROPORTION. 

324.  Four  quantities  are  said  to  be  in  proportion  when  the 
ratio  of  the  first  to  the  second  is  equal  to  the  ratio  of  the 
third  to  the  fourth. 

An  equality  whose  members  are  two  equal  ratios  is  called 
a  proportion.     Thus,  if 

a:b  =  c:  d,  (1) 

then  a,  b,  c,  and  d  are  in  proportion,  or  are  proportional,  and 
equation  (1)  is  2^  proportion. 

A  proportion  can  be  written  in  the  form 

a/h  =  c/d,  a:b  =  c:  d,  or  a:b  : :  c:  d, 

each  of  which  is  read  ^  a  by  6  is  equal  to  c  by  d,'  or  ^  a  is  to 
6  as  c  is  to  d.' 

The  four  numbers  in  a  proportion  are  called  the  propor- 
tionals, or  the  terms,  of  the  proportion. 

The  first  and  fourth  terms  are  called  the  extremes,  and 
the  second  and  third  the  means. 

E.g.,  a  and  d  are  the  extremes,  and  b  and  c  are  the  means  in  the 
proportion 

a  :h  =  c  :  d. 

In  (1),  d  is  called  the  fourth  proportional  to  a,  b,  and  c. 


RATIO  AND  PROPORTION  335 

325.  The  following  theorem  aud  its  converse  in  the  next 
article  are  the  two  fundamental  principles  in  proportion. 

In  any  propoHion  the  product  of  the  extremes  is  equal  to  the 
product  of  the  means. 

That  is,  if  a\b  =  c\d,  (1) 

then  ad  —  be.  (^2) 

Proof.     Clearing  (1)  of  fractions,  we  obtain  (2). 

Ex.   The  first,  second,  and  fourth  terms  of  a  proportion  are  c^,  2  a, 
and  5  h  respectively  ;  find  the  third  term. 
Let  x  =  the  third  term  of  the  proportion ; 

then  c2  :  2  a  =  X  :  5  6. 

.-.  2  ax  =  5  6c2,  or  X  =  5  6cV(2  a). 

326.  Conversely,  if  the  product  of  one  set  of  tivo  numbers 
is  equal  to  the  product  of  another  set  of  two  numbers,  either  set 
can  be  made  the  extremes  and  the  other  set  the  means  of  a 
proportion. 

Proof.     Let  ad  =  bc.  (1) 

Divide  (l)hj  db,  a:b  =  c:d,  ot  c:  d  =  a:b. 

Divide  (1)  by  dc,  a:  c  =  b  :  d,  or  b  :  d  =  a:  c. 

Divide  (1)  by  a6,  d:b  =  c:  a,  or  c:  a  —  d\b. 

Divide  (1)  by  ac,  d  :  c  =  6  :  a,  or  6  :  a  =  d  :  c. 

From  this  principle  it  follows  that  — 

(i)  A  proportion  is  proved  ivhen  it  is  proved  that  the  product 
of  its  extremes  is  equal  to  the  product  of  its  means. 

(ii)  In  a  given  proportion,  we  can  interchange  the  means, 
or  the  extremes,  or  we  can  take  the  means  as  extremes  and  the 
extremes  as  means. 


§172 
6,91 


327.   If 

a:  b  =  c:  d, 

then 

ma  :  mb  =  no  :  nd, 

and 

ma  :  nb  =  mo :  nd. 

336  '  ELEMENTS   OF  ALGEBRA 

328.  Any  proportion,  as       a  :  6  =  c  :  (/,  (1) 

can  he  taken  by 

(i)   inversion  ;  that  is,          b  :  a  =  d :  c,  (2) 

(ii)    alternation  ;  that  is,      a:  c  —  b  .  d,  (3) 

(ill)    addition  ;  that  is,       a  -\-  b  .  a  =  c  -\-  d .  c,  (4) 

or                                             a  -{-  b :  b  =  c  +  d :  d,  (5) 

(iv)    subtraction;  that  is,  a  —  b  :  a  =  c  —  d :  c,  (6) 

or                                              a  -  b :  b  =  c  -  d :  d,  (7) 
(v)    addition  and  subtraction  ;  that  is, 

a  +  b:  a-b  =  c  +  d\c-d.  (8) 

Proof.     From  (1),            ad  =  he.  (1') 

Add  hd  to  (1'),        (a  -f  6) c/  =  (c  +  d)  h.  (2') 

Add  -  hd  to  (1'),   (a  -  ?>)  d  =  (c  -  d)  h.  (3') 
By  §  326,  from  (1'),  we  have  (2)  and  (3);  from  (2'),  (5); 

and  from  (3'),  (7). 

Dividing  (2')  by  (1'),  we  obtain  (4). 
Dividing  (3')  by  (1'),  we  obtain  (6). 
Dividing  (2')  by  (3'),  we  obtain  (8). 
Observe  that  (2)  and  (3)  can  be  obtained  from  (1)  by  (ii)  of  §  326. 

329.  The  products  or  the  quotients  of  the  corresponding 
terms  of  two  proportions  are  x^^^ojyortional. 

That  is,  if                     a:  b  =  c:  d,  (1) 

and                                  a':b'  =  c''.  d\  (2) 

then                             aa^ :  66'  =  c&  :  d&,  (3) 

and  ala!  :  bjb^  =  c/c' :  d/d'.  (4) 

Proof     Multiplying  (1)  by  (2),  by  §§  6  and  91  we  ob- 
tain (3). 

Dividing  (1)  by  (2),  since  ^  =  ^,  we  obtain  (4). 

a  10      010 

330.  Like  powers  or  like  principal  roots  of  proportionals  are 
proportional. 


RATIO  AND  PROPORTION  337 

That  is,  if  a:b  =  c:d,  (1) 

then  a"  :  6"  =  c"  :  d",  (2) 

and  {/a:^b=^c:^d.  (3) 

Proof.     By  §§  128  and  186,  from  (1)  we  obtain  (2). 
Hy  §§  221  and  225,  from  (1)  we  obtain  (3). 

331.  In  a  series  of  equal  ratios  the  sum  of  the  antecedents 
is  to  the  sum  of  the  consequents  as  any  one  antecedent  is  to  its 
consequent. 

That  is,  if  a:  b  =  c  d  =  e  .f=   ",  (1) 

then  a  +  c  +  e  -{-•'  .b  +  d  ^f+   •'  =  a:b  =  c:d=  •.-. 

Proof     Let  a/b  =  r;  then  c/d  =  r,  e//=  ?',  ••• ; 
hence  a  =  br,  c  =  dr,  e  =fr,  •••. 

Adding  the  members  of  these  equations,  by  §  6,  we  obtain 

a  +  c  +  e -f- •••  =  (&  +  <« +/+•••)  »•• 
a4-  c  4-  e  +  •••  _    _a_c_ 
**  b-hd-\-f-\--'~~^~b~d         • 

332.  A  general  and  easy  method  for  proving  a  proportion 
is  to  represent  the  value  of  one  of  the  equal  ratios  in  the 
given  proportion  by  a  single  letter,  as  was  done  in  the  last 
section. 

Ex.  1.    Given  a:b  =  c:d,  prove  that 

a2  +  a6  :  c2  +  cd  =  62  -  2  a&  :  (P  _  2  cd.  (1) 

Let  a/b  =  r  ;  then  c/d  =  r ; 

then  a  =  br,  and  c  =  dr. 

Substituting  these  values  of  a  and  c  in  each  ratio  of  (1),  we  have 

a^  +  ab^  b^r^  -h  6^r  ^  b^(r^  +  r)  ^  &« 
c2  +  cd     d^r^  +  dh'     d%r^  +  r)     d^' 
and  b^-2ab  ^  6^-2  fc^r  ^  b'^(l-2r)  ^  6^ 

d2-2cd      d^-2d^r     d\l-2r)      <P* 
Hence  the  ratios  in  (1)  are  equal. 


338  ELEMENTS   OF  ALGEBRA 

Ex.  2.   Given  a  :  &  =  c  :  d  =  e  :/,  prove  that 

a^  +  c^-^  e^'.h^  +  d^+P  =  ace:'bdf.  (1) 

Let  a/h  =  r  ;  then  c/d  =  r,  and  e//=  r. 

Hence  a  =  br,  c  =  dr,  and  e  =  fr. 

Substituting  the  values  of  «,  c,  and  e  first  in  the  product  of  the 
extremes,  and  then  in  the  product  of  the  means,  we  obtain 

(«3  +  c3  +  e^)bdf  =  (&3  +  #  ^fs^r^Mf, 

and  (63  +  (?3  +  j3)«ce  =  (b^  +  #  j^  p)'fiMf. 

That  is,  the  product  of  the  extremes  in  (1)  is  equal  to  the  product 
of  the  means ;  hence,  by  (1)  of  §  326,  (1)  is  proved. 

333.  A  continued  proportion  is  a  proportion  in  which,  the 
consequent  of  each  ratio  is  the  antecedent  of  the  following 
ratio.     Thus  a,  b,  c,  d  -•'  are  in  continued  proportion  if 

a:b  =  b:c  =  c:  d=  •••. 

li  a:b  =  b  :  c,  then  b  is  called  a  mean  proportional  between 
a  and  c,  and  c  is  called  a  third  proportional  to  a  and  b. 

It  a:b  =  b  :  c  =  c:  d,  then  b  and  c  are  called  the  two  mean 
proportionals  between  a  and  d. 

334.  The  mean  proportional  between  two  numbers  is  equal 
to  the  square  root  of  their  product. 

Proof.     If  a:b  =  b:Cf  then  b^  =  ac,  or  6  =  Vac. 

Exercise  121. 

From  each  of  the  following  products  form  four  different 
proportions  and  their  converses  : 

1.    xy  =  mn.        2.    6x3  =  2x9.        3.    a^-b^  =  a^-y\ 

Find  the  fourth  proportional  to  the  three  numbers  : 
4.    a,  ab,  c.  5.    a^,  2ab,  3b^.  6.    a^,  xy,  5  ay^y. 

Find  the  third  proportional  to  the  two  numbers : 

7.    a%ab.  8.    Q?,2iii?.  9.    Sx,%xy.  10.    1,  x. 


RATIO  AND   PROPORTION 

Find  a  mean  proportional  between  the  two  numbers : 

11.  d',b\  13.    12ax',Sa^ 

12.  2a^,  8  a;.  14.    27  a^b%  3  b. 

li  a  :  b  =  c  :  d,  show  that 

15.  ac:bd  =  c-:(l'. 

16.  d':<^  =  a^-b-:c'-(r-. 

17.  2a-\-3c:3a-\-2c  =  2b-{-Sd:3b  +  2d. 

18 .  la-\-  mb  : ;)«  -f  ^6  =  /c  +  md  :  pc  +  ^d. 

19.  a:  a-\-c  =  a-\-b:a-\-b  -\-c  +  d. 

20.  a-  +  a^  -f-  ^'- :  «-  -  «^  +  ^-  =  c^  4-  ccZ  +  d' :  c- -  cd -{■  d?. 


21.  a  +  &  :  c  +  ^Z  =  Va^  +  6' :  Vc2  +  d^. 

22.  V^M=^:  V?+^  =  ^c?T^:-v/^M^. 

23.  a^c  +  (M?  :  6^d  +  ftd^  =  («  +  c)"' :  (6  +  d)\ 


24.  Va"  +  6"  :  ^c"  +  d"  =  </ar  -  b"- :  </(f  —d\ 

25.  If  a :  ^  =  6  :  c,  prove  that  a:c  =  a-  -.  b\ 

26.  If  a  :  6  =  6  :  c  =  c  :  d,  prove  that  a:d  =  a^:b\ 

Let  r  =  a  -4-  6  ;  then  a  =  br,  b  =  cr,  c  =  dr. 
.'.  abc  =  bcdr^.     .-.  a  -^  d  =  r^  =  a^  -^  b^. 

27.  If  a,  6,  c,  (i  be  any  four  numbers,  find  what  numbei 
must  be  added  to  each  to  make  the  results  proportional. 

28.  Two  numbers  are  in  the  ratio  of  3  to  8,  and  the  sum 
of  their  squares  is  3577 ;  find  them. 

29.  The  ages  of  two  persons  are  as  3:4,  and  30  years 
ago  they  were  as  1 :  3 ;  find  their  present  ages. 

30.  The  sides  of  a  triangle  are  as  3  :  4  :  5,  and  the  perim- 
eter is  480  yards ;  find  the  sides. 


340  ELEMENTS   OF  ALGEBRA 

31.  Divide  the  number  14  into  two  such  parts  that  the 
quotient  of  the  greater  divided  by  the  less  shall  be  to  the 
quotient  of  the  less  divided  by  the  greater  as  16  to  9. 

32.  Show  that  the  ratio  of  any  two  fractions,  not  involv- 
ing surds,  can  be  expressed  by  the  ratio  of  two  whole 
numbers. 

33.  Express  the  ratio  of  5^  to  7y\  by  the  ratio  of  two 
whole  numbers. 

34.  Express  the  ratio  of  17 J  to  14|  by  the  ratio  of  two 
whole  numbers. 

35.  The  sum  of  two  numbers  is  8,  and  their  product  is 
to  the  sum  of  their  squares  as  3  to  10.  What  are  the 
numbers  ? 

36.  The  sum  of  two  numbers  is  10,  and  the  sum  of  their 
squares  is  to  the  square  of  their  sum  as  13  to  2^.  What 
are  the  numbers  ? 

37.  A  hare  is  pursued  by  a  greyhound,  and  is  60  of  her 
own  leaps  before  him.  The  hare  takes  3  leaps  in  the  time 
that  the  greyhound  takes  2 ;  but  the  greyhound  goes  as  far 
in  3  leaps  as  the  hare  does  in  7.  In  how  many  leaps  will 
the  greyhound  catch  the  hare  ? 

Let  X  =  the  number  of  leaps  taken  by  the  greyhound, 
and      y  =  the  number  of  leaps  taken  by  the  hare  in  the  same  time  ; 
then  aj :  y  =  2  :  3, 

and  af-f60:y  =  7:3. 


CHAPTER   XXVI 
THEORY  OF  EXPONENTS 

335.  Hitherto  we  have  defined  and  used  only  positive 
integers  as  exponents.  It  is,  however,  found  convenient  to 
extend  the  meaning  of  an  exponent  so  that  we  can  use  zero, 
a  fraction,  or  a  negative  number,  as  an  exponent. 

As  it  is  desirable  that  all  exponents  should  obey  the  same 
laws,  we  shall  fix  the  meaning  of  (i.e.,  define)  any  new  expo- 
nent by  imposing  the  restriction  that  all  exponents  must 
obey  the  fundamental  law, 

a"*  xa"  =  a'"-^".  (1) 

E.g.,  to  find  the  meaning  of  a',  we  have  by  law  (1), 

that  is,  (a' )3  =  a  ;        .*.  a»  =  the  cube  root  of  a.  §  213 

Again,  to  find  the  meaning  of  a',  we  have  by  law  (1), 

J-J  =  J-^^  =  J  =  a^; 
that  is,  (a*)2  =  ««  .        .^  ^t  =  t^e  square  root  of  a\  §  213 

336.  Meaning  of  a  positive  fractional  exponent. 

Let  r  and  s  denote  any  positive  integers;  then  by  the 
fundamental  law  of  exponents,  we  have 

n       ,  ■< 1-  —  to  «  terms 

a' '  a*  •••  to  s  factors  =  a*   * 


that  is,         (a')'  =  a*";     .-.  a'  =  the  sth  root  of  a'' ;  §  213 

r 

that  is,  a'  is  only  another  way  of  writing  the  sth  root  of  the 
rth  power  of  a. 

S41 


342  ELEMENTS   OF  ALGEBRA 

Hence  a  positive  fractional  exponent  denotes  a  root  of  a 
power  of  its  base.     The  denominator  indicates  the  root,  and 

the  numerator  the  power. 
1 
Thus  a'  denotes  the  sth  root  of  a. 

r 

337.  Using  a*  to  denote  only  the  principal  sth  root  of  a*", 
we  have 

r 

a'  =  W={-^a)%  §§220,226 

and  a«  —  ^a. 

Ex.  1.  83=^8  =  2;        4^  =  ( ^4)^  =  2^  =  8. 

Ex.  2.  8^  =  (^8)2  =  22  =  4. 

Ex.3.  (-32)^  =  (\/^r32)4=(_2)4=16. 

338.  Meaning  of  zero  as  exponent. 

By  the  fundamental  law  of  exponents,  we  have 

a"*  •  a^  =  cr+^  =  «'" ; 
.-.  a^  =  qT'/oT'  —  1. 

That  is,  any  base,  except  zero,  with  zero  as  an  exponent  is 
equal  to  positive  one. 

Observe  that  a°  is  only  another  way  of  writing  a'^/a'^,  or  1. 
E.g.,  a'i  =  a/a  =  a'^/a^-  =  a'^/a^="'  =  l. 

339.  Meaning  of  a  negative  exponent. 

Let  n  denote  any  positive  integer  or  fraction;   then  by 
the  fundamental  law  of  exponents,  we  have 

a"  •  a-'^  =  a"+(-'^)  =  a^=l. 
.:  a-"  =  l/a". 

That  is,  a~"  is  only  another  way  of  writing  the  reciprocal 
of  a",  or  of  denoting  that  a'*  is  to  be  used  as  a  divisor. 
E.g.,  3-2  =  1/32  =  1/9  ;  (-  2)-3  =  l/(_  2)3  =  -  1/8. 


THEORY  OF  EXPONENTS  343 

Note.  The  arithmetic  value  of  an  exponent  denotes  a  power,  or 
a  root  of  a  power,  of  its  base ;  and  its  quality  denotes  whether  this 
power  or  root  is  to  be  used  as  a  factor  or  divisor. 

Fractional  and  negative  exponents  express  no  new  ideas,  and  are 
not  necessary  to  the  notation  of  algebra ;  but  they  are  very  con- 
venient, and  greatly  facilitate  many  operations.  Fractional  ex- 
ponents simply  afford  another  way  of  writing  a  root  of  a  power ;  and 
negative  exponents,  another  way  of  writing  a  divisor. 

r 

340.  Hereafter  in  this  chapter  we  shall  use  a'  to  denote 
only  the  pnncipcd  sth  root  of  a*",  or,  what  is  the  same  thing, 
the  rth  power  of  the  principal  sth.  root  of  a. 

Observe  that  r  can  be  either  positive  or  negative,  but 
that  s  is  always  positive. 

Ex.  1.    8-i  =  2-2  =  1/22  =  1/4. 
Ex.  2.    (-  27)-^  =  (-  3)-*  =  l/(-  3)4  =  1/81. 
Ex.3.    (-32)-f  =  (-2)-«  =  l/64. 
Note  the  advantage  of  tirst  extracting  the  root  in  these  examples. 

341.  A  base  with  any  exponent  is  called  an  exponential 
expression  ;  as,  3^,  a',  (x  -f-  ?/)",  (f^'. 

342.  The  quality  of  an  exponent  can  be  changed  if  the 
sign  before  the  exponential  expression  is  changed  from  x 
to  -^,  or  from  ^  to  x . 

Proof.  axb-"  =  ax  (1/6")  =  a  -t-  6".  (1) 

Also  a  -h  b-»  =  a-T-  (1/6")  =  a  x  6".  (2) 

343.  Any  exponential  factor  can  be  transfeired  from  the 
dividend  to  the  divisor,  or  from  the  divisor  to  the  dividend,  if 
the  quality  of  its  exponent  is  changed  from  -^  to  —,  or  from 
-  to  -h. 

Proof     This  is  the  converse  of  (1)  and  (2)  in  §  342. 

Or  this  operation  is  simply  multiplying  both  dividend 
and  divisor  by  the  same  exponential  expression. 


344  ELEMENTS   OF  ALGEBRA 


Ex.  1.    a-Vc2  =  c-2/a2.  a-^/b-^  =  bya^ 
Ex.  2.    — — -  =  — -  =  cH^a^y^  = 


Exercise  122. 
Eind  the  value  of  each  of  the  following  expressions : 

1.  4-1  5.   (2/3)-^        9.  8t  13.  (27/8)1 

2.  5-^  6.  1/5-3.         ^Q    4-f.  14.   (81/16)"l 

3.  (3/4)-\       7.  l/.Z-\        11.  9-1  15.  (-27)1 

4.  (2i)-2.         8.  35".  12.  (4/9)-l      16.  (-  125)-^ 

17.    8-^.4-2.     18.    (l/64)-3  .  (l/9)i      19.    (1/25)"^  .  27~l 

Write  each  of  the  following  expressions  without  using 
fractional  or  negative  exponents : 

20.  ah.  22.  3i»i  24.  2ax~^.        26.  {x/y)^. 

21.  x^.  23.  2a;V^-       25.   (a/6)i       27.  (a;/?/)  ". 
28.    ^—  31.    ^.  34.      "-^^'"  . 

5  a-%^  x-^y-h-^  (a  +  hyx-^y 

'    lyh-'  '     ab-'c-'  '       (a-\-b)-' 

^^     5x-'y-h\         ^^     b(x-y)-\         ^^     x--(x  +  y)-^ 


7  c-'db- 


(x-\-y)-^  a--{x  +  yy' 


37.  Write  each  of  the  expressions  in  examples  28  to  36 
in  the  integral  form,  i.e.y  transfer  all  the  factors  from  each 
denominator  to  the  numerator. 

38.  Using  fractional  exponents,  write  V^^5  -\fa^''>  '\/^\ 

39.  Using  negative  fractional  exponents,  write  Vi/^; 

^17^;  -^17^';  ViT^. 


THEORY  OF  EXPONENTS  346 

344.  The  meaning  of  any  real  commensurable  exponent 
having  been  determined,  it  remains  to  prove  that  any  such 
exponent  obeys  all  the  laws  of  positive  integral  exponents. 

For  convenience  in  stating  these  laws  we  shall  enlarge  the 
meaning  of  the  word  power  so  as  to  include  whatever  is 
denoted  by  an  exponent. 

345.  Exponents  were  so  defined  as  to  obey  the  following 
fundamental  law : 

Tlie  product  of  the  mth  and  the  nth  power  of  any  ba^e  is 
equal  to  the  (m  +  ri)th  power  of  that  Inxse. 

That  is,  a^^a"  =  «*"+",  I 

where  m  and  n  denote  any  commensurable  real  numbers. 

Ex.  1.   x^  .  a;^  =  x^'^^  =  x^. 

Ex.  2.   x~^  .  a;"^  =  x~^^"^)  =  x~^. 

Ex.  3.   x^a~^  '  x-hi^  =  x-^a~^  =  l/(xa^). 

346.  Tfie  quotient  of  the  mth  power  of  any  base  divided  by 
the  nth  power  of  the  same  base  is  equal  to  the  (m  —  n)th  power 
of  that  base. 

That  is,  a'^/a"  =  a'"-",  II 

where  m  and  n  denote  any  commensurable  numbers. 

Proof  a*"  H-  a"  =  a"*  x  a""  =  a"-".  §§  342,  345 

Ex.  1.   x-8  ^  X-*  =  x-3-(-'«)  =  X.  §§  342,  345 

Ex.  2.   J-^  oT^  =  a^-(-^)  =  al 
Ex.  3.   a~^  -r  a"^  =  a~^~^~^^  =  a^. 

347.  By  §251,  V^=Va^*;   .-.  a'  =  a^. 

r^p  r»^p8  rq+pt 

Hence  a'  '  =  a*«  *«  =  a  '"  . 

Ex.  1.  x^ .  x^  =  x^*  =  x^2"  =  x"^. 
Ex.  2.   x^/x^  =  x*"^  =  x^  =  xT^. 


846  ELEMENTS  OF  ALGEBRA 

Exercise  123. 
Simplify  each  of  the  following  expressions : 

1.  x'x\  5.    x^3fx-\  9.    a^6-2  X  a-^h\ 

2.  a-W.  6.    a-^a-V.  10.    ^  x~^^y^  x  2  x~^y^ . 

3.  rt'V.  7.    «y.  11.    4  a;~^a~^  X  3  a^' ai 

4.  a;-2a;-l  8.    x'^^-x^.  12.    2A^x5:K"V^r. 

13.  a/a-\       15.    4-74-^        17.    a^-yiu-^        19.    x-'^/x-l 

14.  .'^Va;-^         6.    7-77-^        18.    x-'/x-^\       20.    t/V^-I 

21.  (8x--)/(6a.-0.  29.    ^-^^"1 

22.  .^'"-70?-"*. 


30.  ic^/a;  ^. 

31.  a^^^.aM. 


23.  x-^/x^-'^. 

24.  ar'-ya?2-n^ 

25.  x''^/y?-'\  32.    a'^aj"^^  •  a^ici 


26.  o^-Yr--.  33^    a.V/(^-Vb. 

27.  a.-^-a^"l  34.    a/^^^  v'^^. 

28.  aWa"^.  35.    .J/a?'^  •  ^ic'" -j- a^^. 

348.   If  m  and  n  denote  any  commensurable  real  numbers, 
the  other  three  laws  of  exponents  are : 

The  nth  poicer  of  the  mth  power  of  any  base  is  equal  to  the 
mnth  power  of  that  base. 

That  is,  (a'")"  =  a'"".  Ill 

The  nth  power  of  the  product  of  any  number  of  factors  is 
equal  to  the  product  of  the  nth  powers  of  the  factors. 

That  is,  (ab.'.y  =  a"b"  '".  IV 


THEORY  OF  EXPONENTS  347 

TJie  nth  power  of  the  quotient  of  one  number  by  another  is 
equal  to  the  quotient  of  the  nth  power  of  the  first  by  the  nth 
power  of  the  second. 

That  is,  {a/by  =  a"/b".  V 

Ex.  1.  (a:2)-3  =  x2(-3)  =  x-e  =  l/x*.  By  IH 

Ex.  2.  (8-2)-i  =  8-2H)  =  8^  =  16.  By  HI 

Ex.  3.  (a"^)~^  =  a~^^"^^  =  a^.  By  III 

Ex.  4.  (x-iy-8)-2  =  (a;-i)-2(y-8)-2  =  x^^.  By  III,  IV 

Ex.  5.  (9"^  x-6)"^  =  (9"*)~^(x-6)"^  =  9^  X*  =  3  X*. 

Ex.  6.  (x2/a-8)-2  =  (a;2a3)-2  =  a;-*a-«. 

Ex.  7.  {x^/y^y^  =  (x^y"^)-»  =  x'V. 

Ex.8.  (^_£l\-''^(l^\-'^^:!^ 

\^y'l  \zy^l         3-S2,-t 

=  27  2/^x8/8. 

Ex  9    Q^v^^        rT>/R_aV^  .   /a6-2\^ 
6\/a^   '^^v^      6a"^   '  \ba-^l 

=  aV^^(a26-8)^ 

349.  Proof  of  laws  III,  IV,  V  when  m  and  n  are  positive 
fractions. 

Let  p,  g,  r,  and  s  denote  any  positive  integers. 

To  prove  III,           (J)'  =  (V^)'  §  337 

=  (V^y  §  227 

=  [(V«W  §226 

=  (^a)^  =  a".  Ill 


348  ELEMENTS   OF  ALGEBRA 

r 

To  prove  IV,  (aby  =  -</{aby  §  337 

=  ^tf^  §  119 

=  </a''<JW  §  224 

r  r 

=  a'b\  IV 


To  prove  V,  (a/by  =  </(a/by  §  337 

=  -</ar/b''  §  186 

=  </a^/</F        -       '  §225 

r        r 

=  a'/b^.  V 

350.   Proof  of  laws  III,  IV,  V  wJien  m  and  n  are  negative. 

Let  h  and  k  denote  any  positive  integers  or  fractions. 

To  prove  III,      (a"^)"*  =  1  --  (1/a*)*  §  339 

=  1 -f- (l/a'^*)  §§  186,118 

=  a^\  III 

To  prove  IV,         (ab)-^  =  l/{aby  §  339 

=  l/a'^^*  §  119 

=  (1/a*)  (1/6*)  =  a-^b-\  IV 

To  prove  V,         (a/b)-^  =  1  --  {a/by  §  339 

=  1  -f-  (ay^'^)  §  186 

=  by  a''  =  a-'^/b-K  V 

The  verification  of  laws  III,  IV,  and  V  when  m  or  n  is 
zero  is  left  as  an  exercise  for  the  pupil. 

Exercise  124. 
Simplify  each  of  the  following  expressions : 

!•    (O'-  4.    {x-^\  1-    («-")-*• 

2-    W"'-  5.    {x-i)-i.  8.    {aSl 

3.    (a;')-*.  6.    («")-«  9.    (-5^*^^'. 


THEORY  OF  EXPONENTS  349 

10.  (VaryK  14.    (4.a-yK  18.    (x~^a^)-^^. 

11.  (a/^)-'*.  15.    (Sa'')i  19.    (a-2c-t)-8. 

12.  (VO'"  ^^-    (^"'2/"')"'-  20.    (a-y32)-i. 

13.  (a;V2)-3.  17.    (8a~V^)"^.        21.    (2a-Vaj)"' 

22.  (3ar^^2)-l  33.    </^^yV^. 

23.  (aj-"c-)"'.  34.    ^(a  +  &)'  •  (a  +  bp. 

24.  (3v^^o-«.  35.  f^^Y^f^^y. 

25.  (ay.-)-.^  y^"^        ,^«"^'^^ 

26.  [-2-^a-y(4a;-i)]-2.  36.       J^./^^Y^^ 

27.  (32a-i/^)-i  U^-U-.-i 

28.  ^^«/^.  37.    (1^     .(^^  '. 

29.  V^?6^/^5/S^.  ^„  »^-|      ^„-ii,h^- 

38 


30.  V^rY^'/VsrY^\ 

31.  V^Fy^^=^.  ^^Y^«/^ 

32.    ^(a  +  6)«.(a2-62)-i.  *    V^-^a/    "  Xa;-^* 

351.  The  following  examples  are  applications  of  the 
methods  of  multiplication,  division,  and  evolution,  to  poly- 
nomials whose  terms  involve  fractional  and  negative  expo- 
nents : 

Ex.  1.     Multiply  >  +  1  +  1/  ^a  by  ^a  +  1/  ^a  -  1. 

The  terms  +  1  and  —  1  may  be  regarded  as  the  coefllcients  of  a^ ; 
hence  arranging  both  expressions  in  descending  powers  of  a,  we  have 

a^  +  1  +  a~^ 
g^  -  1  +  q~^ 
a^  +  a^  +  1 

-  a3  -  1  -  a  3 

+  1  +  a~^  +  g"^ 

af  +1  +  a~^ 


350 


ELEMENTS  OF  ALGEBRA 


Ex.  2.     Divide  16  a-^  +  5  a-i  -  6  a-2  +  6  by  1  +  2  a-K 

Arrange  in  ascending  powers  of  a. 

16  a-3  -    6  a-2  4-  5  a-i  +  6  [  2  «-i  +  1 
16a-3+    8q-2  8  a'^  -  7  a"!  +  6 

-  14  a-2  +  5  a-i 

-  14  a-2  -  7  g-i 

12  a-i  +  6 
12a-i  +  6 

Ex.  3.     Eind  the  square  root  of 

4  X  +  2  x^'  -  4  x^  -  4  x^  +  x^  +  xi 
Arrange  in  descending  powers  of  x. 

x^  -  4 x^  +  2 x^  +  4 X  -  4x^  4-  x^  |  x^  -  2 x^  +  x «" 

J 

2  x^  -  2  x^ 


xt 

-4x^  +  2x^  +  4x 
-4x^              +4x 

-4 

x^  +  x^ 

2x^          -4x^  +  x^ 
2x«           -4x^  +  x^ 

Exercise  125. 


Multiply : 

1.  a^-\-ah^-^b^  by  a^-bi- 

2.  x^  —  ic^2/^  +  y^  by  a?^  +  y^. 

3.  3aj^  — 5  + 8a;~^  by  4 a;^  + 3 aj~i 

4.  3a^-4a*-a~^  by  3 a^ -^ oT^ -  6 oT^ 

5.  a^  +  a^6^  +  6^  by  a^  —  ah^-{-bK 

6.  ic^  —  a;2  +  a?^  —  aj  by  a;^  +  x^. 

5  3  1  -1    1  3  1 

7.  a;^  —  a?^  +  a?^  —  a?  *  by  a;^  +  ^^. 

8.  a3  +  &^  4-  c^  —  bh^  —  c^a^  —  a^6^  by  a^  +  b^  -\-  ci 


THEORY  OF  EXPONENTS  351 

n  n 

9.    x'^  +  x^-j-l  by  X-" -\- x~~^ -}- 1. 

10.  iJ-^i^a^^i-^i^ah-^b^  by  ia^  +  ^bi 

11.  (f +  2c-'-7  by  5-3c-'=  +  2(f. 
Divide : 

12.  21x  +  a;^4-a;^4-l  by  3a;^4-l. 

13.  15a-3a^-2a"*  +  8a-i  by  5a*4-4. 

14.  55^-66^-4fe-^-46~*-5  by  6^-26"^. 

15.  21a'^  +  20-27a^-26a2'  by  3a*-5. 

16.  a;^y~^  +  2  H-  ic~^y^  by  x^2/~^  —  1  +  x~^yK 

17.  a^  +  a-6^  —  ah^  —  ab  +  ah^  +  &^  by  a^  4-  &i 

18.  x^y~^  +  y^x~^  by  a;'?/"^  +  y^a;~». 

19.  «'  —  2  -h  a~3  by  a^  —  a '. 

20.  8  c-"  -  8  C  +  5  c^"  -  3  c-3"  by  5  c"  -  3  c"". 
Find  the  square  root  of  the  following  expressions : 

21.  25a*  +  16-30a-24a.^  +  49a*. 

22.  9x-12x^-\-10-4:X~^-{-x-\ 

23.  4  ay^a-'  +  12  a;a-^  +  25  +  24  x-^a  + 16  aj-^a^ 

24.  25  arV'  +  i  y^x-^  -  20  ic^/"^  -  ^yx'^  +  9. 

25 .  x^  —  2  oT^x^  +  2  a^ic^  +  a~h^  —  2  a^a;^  +  ai 

26.  4  .T"  +  9  «-"  +  28  -  24  aj-i"  - 16  a;K 

27 .  9  a;-^  -  18  a;- Vy  +  15  2/  -^  a^  -  6  Vp  ^  a;  +  ^V^. 


352  ELEMENTS   OF  ALGEBRA 

352.  The  following  examples  are  applications  of  the  for- 
mulas for  products  and  quotients  in  Chapter  IX.,  to  binomials 
whose  terms  involve  fractional  and  negative  exponents  : 

(1)  (2  a^  -  x~i)^  =  (2  Jy  +  3(2  a^)2(-  x~^) 

+S(2  J)(-x~^y+(-a~^y.  (1) 
=  8  a^  -  12  ax~^  +  6  a^x-i  -  a"i         (2) 

(2)  (x^  +  yi)  (x^  -  yi)  =  {x^^  -  {y^y  =  x^-  yK  §  122 

(3)  (7  x-9  2/-1)  (7  x+9  y-i)  =49  a;2_8l  y-\ 

(4)  (4a;-5a;-i)(4x+3x-i)  =  16a:2+(3x-i-5x-i)4ic-15ic-2       §123 

=  16  x2  -  8  -  15  a;-2. 

(5)  (pfi  -  1)  -  (X^  -  1)  =  [(X^)5  -  15]  ^  (x^  _  1) 

=  (x^f  +  (x^s  ^  (a;^-)2  +  x^  +  1  §  129 

=  a;'^  +  ic  +  cc^  +  ic^  +  1. 

(6)  (x^  +  27)  -  (x^  -t-  3)  =  i(xh^  +  33]  -  {x^  +  3) 

=  X  -  3  a:^  +  9. 

Exercise  126. 
Write  the  value  of  the  following  expressions : 

1.  {ar^-\-h^f.  3.    {m^  +  Jf.  5.    (f^  -  s'^f. 

2.  (x^^y-^f,  4.    (c2-6*)l  6.    (2a^-a-^f. 

7.  (r-2  +  6^y.  11.    (a^  +  &~^/. 

8.  {i^'-^n-^y.  12.    (a^^-2/"^)«. 

9.  (iVa-i-^6)^  13.    (o?^  +  1)  (aJ^  -  1). 

10.     (a~^-2  62c^)^  14.     (x^-\.y^)(x^-y^), 

15.  (4  a;^  +  3  a"^)  (4  a;^  -  3  a"*). 

16.  (3  a;  -  5  a-^  (3  a;  +  2  a-i). 


THEORY  OF  EXPONENTS  353 

17.  (ab-c^(ab-^5x-^). 

18.  (x-9a)-^{x^  +  3a^). 

19.  (a-2x_i6)^(a-x_4). 

20.  (aj-3- +  8) -f- (a;-*  4- 2). 

21.  (c2*  -  c-'') -=- (c*  -  c-i*). 

22.  (l-8a-3)^(l-2a-^). 

23.  (a;*  +  ic*  +  1)  (a;^  -  1). 

24.  (a;-'*-l)^(a;-i  +  l). 

25.  (ar^+-32)H-(a;'*  +  2). 

26.  (a,-3  -  2/S)  ^  (a;^  —  2/^). 

27.  (ar^  +  /)^(a;*  +  A 

28.  (a;  -  243  2/^)  ^(a;^- 3  2/^. 


CHAPTER   XXVII 
INDETERMINATE  EQUATIONS  AND  SYSTEMS 

353.  Division  by  zero.  As  a  quotient,  0/0  denotes  the 
number  which  multiplied  by  0  is  equal  to  0  (§  85).  By 
§  74  any  number  multiplied  by  0  is  equal  to  0 ;  hence  0/0 
denotes  any  number  whatever,  or  is  indeterminate.  That  is, 
when  the  dividend  is  zero,  division  by  zero  is  indeterminate. 

As  a  quotient,  a/0  denotes  the  number  which  multiplied 
by  zero  is  equal  to  a.  But  any  number,  however  large, 
multiplied  by  zero,  is  zero;  hence  the  division  of  a  by  0^ 
is  impossible.  That  is,  ivhen  the  dividend  is  not  zero,  division 
by  zero  is  impossible  in  the  sense  that  no  number  can  express 
the  quotient  or  any  part  of  it. 

354.  The  forms  0/0  and  a/0.  As  an  answer  to  a  problem 
the  indeterminate  form  0/0  denotes  that  the  problem  is 
indeterminate,  i.e.,  has  an  unlimited  number  of  answers. 

As  an  answer  to  a  problem,  a/0  denotes  that  the  problem 
involves  inconsistent  conditions,  and  is  therefore  impossible, 
as  is  illustrated  by  the  following  problem  : 

Prob.  A  and  B  are  travelling  in  the  direction  PB  at  the  rates  of  a 
and  h  miles  per  hour.  At  12  o'clock  A  is  at  P  and  B  at  ^,  which  is  c 
miles  to  the  right  of  P.     Eind  when  they  are  together. 

P Q B 

Let  distances  measured  to  the  right  from  P,  and  periods  of  time 
after  12  o'clock,  be  regarded  as  positive. 

Let  X  =  the  number  of  hours  from  12  o'clock  to  the  time  when  A 
and  B  are  together. 

Then  ax  =  6x  +  c.  (1) 

Hence  x  =  — —•  (2) 

•  a  —  0 

364 


INDETERMINATE  EQUATIONS  AND   SYSTEMS     355 

Discussion.  If  c  >  0  and  a  >  6,  a;  is  positive  ;  that  is,  A  will  over- 
take B  at  some  time  after  12  o'clock. 

If  c  >  0  and  a  <  &,  x  is  negative  ;  that  is,  A  and  B  were  together 
at  some  time  before  12  o'clock. 

If  c  =  0  and  a  =^  b^  x  =  0;  that  is,  A  and  B  are  together  at  12 
o'clock,  but  not  before  or  after  that  time. 

If  c  =  0  and  a  =  b,  x  =  0/0  ;  that  is,  A  and  B  are  always  together 
under  the  conditions;  and  the  problem  is  indeterminate,  i.e.,  has  an 
unlimited  number  of  answers. 

If  c  ^fc  0  and  a  =  6,  a;  =  c/0 ;  that  is,  A  and  B  can  never  be 
together  as  they  are  always  at  a  fixed  distance  apart ;  the  problem 
involves  inconsistent  conditions,  and  is  therefore  impossible. 

Observe  that  the  fraction  — - —  assumes  the  form  -  by 

a-b  0    -^ 

reason  of  two  indej^endent  conditions;  namely,  c  =  0  and 
a  =  b.  In  any  such  case  the  form  0/0  indicates  that  the 
given  fraction  can  have  any  value  under  the  conditions. 

355.  An  impossible  equation  is  one  which  expresses  a  con- 
dition which  cannot  be  satisfied. 

E.g.,  Sx  +  5  =  Sx  —  S  is  an  impossible  etjuation ;  for  it  expresses 
the  condition  0  •  a;  =  —  13,  which  no  value  of  x  can  satisfy. 

Again,  ^a;  =  —  3  is  an  impossible  equation,  when  ^x  is  restricted 
to  its  principal  value. 

An  impossible  system  of  equations  is  a  system  whose  equa- 
tions are  inconsistent  (§  206). 


E.g.,  the  system  ax  -{-  by  =  c,     (1) 

3  ax  +  3  6y  =  5  c,  (2) 


}   («) 


is  impossible  ;  for  its  equations  are  evidently  inconsistent  (§  206). 

An  impossible  equation  or  system  of  equations  is  often  a  particular 
case  of  a  more  general  equation  or  system,  in  which  the  solutions 
involve  the  form  a/0. 

Thus,  the  equation  ax  =  b  becomes  impossible  only  when  a  =  0, 
and  then  its  root  b/a  becomes  6/0. 

It  will  be  seen  in  §  356  that  a  system  of  two  linear  equations  in 
X  and  y  becomes  impossible  only  for  a  certain  relation  between  the 
coefficients  of  its  equations,  which  makes  the  values  of  x  and  y  assume 
the  form  a/0. 


856  ELEMENTS   OF  ALGEBBA 

Again  the  system 

x  +  y  =  9,  .      (1) 

2x  +  y  =  lS,  (2)  [  (&) 

x  +  5y  =  16,  (3). 

is  impossible ;  for  the  only  solution  common  to  (1)  and  (2)  is  4,  5, 
and  this  reduces  (3)  to  29  =  16. 

Equation  (3)  cannot  be  obtained  from  (1)  or  (2),  or  by  combining 
(1)  and  (2);  hence  it  is  independent  of  them  separately  and  jointly. 

System  (b)  illustrates  the  principle  that 

When  the  number  of  independent  equations  in  a  system  ea>- 
ceeds  the  number  of  unknowns,  the  system  is  impossible. 

356.  A  defective  system  is  one  which,  lacks  one  or  more  of 
the  full  number  of  solutions  which  we  would  expect  from 
the  degrees  of  its  equations. 

E.g.^  the  system 


|(«) 


a2x2  -  62y2  =  c2,  (1) 

ax-(h-\-  e)y  =  c,  (2) 

which  has,  in  general,  two  solutions  (§306),  becomes  defective  when 

e  =  0. 

For,  dividing  (1)  by  (2)  when  e  =  0,  we  obtain 

ax  +  by  =  c.  (3) 

Equations  (2)  and  (3)  form  a  system  equivalent  to  system  (a); 
hence  system  (a)  has  but  one  solution  when  e  =  0. 

357.  An  indeterminate  equation  is  one  which  has  an  un^ 
limited  number  of  solutions.  Thus  any  equation  in  two  or 
more  unknowns  is  indeterminate. 

An  indeterminate  system  of  equations  is  one  which  has  an 
unlimited  number  of  solutions. 


E.g. J  the  system  Sx  +  4:y-{-5z  =  0, 

x-y-2z  =  0,  ^^^^ 


Ida] 


is  an  indeterminate  system  ;  for,  assigning  any  value  whatever  to  z, 
we  can  find  a  corresponding  set  of  values  of  x  and  y.  Hence,  system 
(a)  has  an  unlimited  number  of  solutions,  and  is  indeterminate. 


INDETERMINATE  EQUATIONS  AND   SYSTEMS     357 

Again,  the  system 

2x-\-Sy-z  =  lo,  (1) 

Sx-y  +  2z  =  8,  (2) 

5  X  H-  2  y  +  2!  =  23,  (3) 

is  indeterminate.  No  two  of  its  equations  are  equivalent,  but  any 
one  of  them  can  be  obtained  from  the  other  two  ;  thus,  by  adding 
(1)  and  (2),  we  obtain  (3).  Hence  the  system  contains  but  two  in- 
dependent equations,  and  therefore  any  solution  of  two  of  them  will 
be  a  solution  of  the  third. 

These  examples  illustrate  the  following  principle  : 

WJien  the  number  of  independent  equations  in  a  system  is 
less  than  the  number  of  unknowns,  the  system  is  indeterminate. 


Ex.   By  discussing  its  solution,  show  that  tlie  system 
ax-i-  by  =  c, 
a'x  +  b'y  =  c', 


|(«) 


is  (i)  indeterminate  if  ^  =  _  =  2_  j  (1^ 

a'     b'     c' 

and  (u')  impossible  if  ^  =  ^  r,^  ^.  (2) 

a      b'      c' 

By  §  207  the  values  of  x  and  y  in  system  (a)  are 

_  b'c  —  be'         _  ac'  —  a'c  .on 

~  ab'  -  a'b     ^     ab'  -  a'b  ^  ^ 

(i)  When  condition  (1)  is  satisfied,  from  (1)  we  have 

ab'  -  a'b  =  0,   b'c  -  be'  =  0,  ac'  -  a'c  =  0  ; 

hence  the  values  of  x  and  y  in  (3)  each  assume  the  form  0/0  ;  that  is, 
the  system  has  an  unlimited  number  of  solutions,  and  is  therefore 
indeterminate. 

(ii)  When  condition  (2)  is  satisfied,  we  have  ab'  —  a'b  =  0. 

But  neither  b'c  —  be'  nor  ac'  —  a'c  is  zero. 

Hence  the  value  of  each  x  and  y  assumes  the  form  a/0 ;  that  is, 
the  system  has  no  solution,  and  is  therefore  impossible. 

The  equations  in  (a)  are  evidently  equivalent  when  (1)  is  satisfied, 
and  inconsistent  when  (2)  is  satisfied. 


358  ELEMENTS   OF  ALGEBEA 

368.  Sometimes  it  is  required  to  find  the  positive  integral 
solutions  of  an  indeterminate  equation  or  system. 

The  following  examples  will  illustrate  the  simplest  general 
method  of  finding  such  solutions. 

Ex.  1.    Solve  7  X  +  12  y  =  220  in  positive  integers. 

Dividing  by  7,  the  smaller  coefficient,  expressing  improper  fractions 
as  mixed  numbers,  and  adding  the  proper  fractions,  we  obtain 

aj  +  y  +  ^pi=:31.  (1) 

Since  x  and  y  are  integers,  31  —  x  —  y  is  an  integer ;  hence  the 
fraction  in  (1)  denotes  an  integer. 

Multiplying  this  fraction  by  such  a  number  as  will  make  the  coeffi- 
cient of  y  divisible  by  the  denominator  with  remainder  1  (which  in  this 
case  is  3) ,  we  have 

- — ^ =  2  ?/  —  1  +  ^—^ —  =  an  mteger. 

Hence  ^ =  an  integer  =  p,  suppose. 

:.y  =  lp  +  1.  .     (2) 

From  (1)  and  (2),      x  =  28-12j9.  (3) 

Since  x  and  y  are  positive  integers,  from  (2)  it  follows  that^  >—  Ij 
and  from  (3)  it  follows  that  p  <  3  ;  hence 

i)  =  0,1,2.  (4) 

From  (2),  (3),  and  (4),  we  obtain  the  three  solutions 

a:  =  28,  16,     4; 

y=    2,     9,  16. 

Ex.  2.   Solve  in  positive  integers  the  system 

a;  +  2/  +  ^  =  43,  (1) 

10  aj  +  5  ?/  +  2  ;s  =  229.  (2)  J 


INDETERMINATE  EQUATIONS  AND  SYSTEMS     359 

Eliminate  z,  8x-\-Sy  =  143, 

y  +  2x-\-^-^^  =  ^7.  (3) 

.-.  i^LzA  =  x-l+  ?-=-i  =  an  integer. 

o  o 

x—1 

.'. =  an  integer  =  p,  suppose. 

o 

.-.  x  =  3p  +  l.  (4) 

From  (3)  and  (4),  y  =  io-8p.  (6) 

From  (1),  (4),  and  (5),      z  =  5p-  3.  (6) 

From  (6),  i)  >  0  ;  and  from  (o),  />  <  6  ;  hence 

p=    1,     2,     3,     4,     5.      . 
Whence  x  =   i,     7,  10,  13,  16; 

y  =  37,  29,  21,  13,     5  ; 

z=    2,     7,  12,  17,  22. 
Thus,  the  system  has  five  positive  integral  solutions. 


Exercise  127. 
Solve  in  positive  integers : 

1.  3a; +  291/ =  151.  8.    12 x -lly  +  4:Z  =  22, 

2.  3.'c  +  82/  =  103.  -4:X-{-5y-\-z  =  17. 

3.  7x-^12y  =  152.  9.    20a:  -  21?/ =  38,  1 

4.  13a;  +  72/  =  408.  S y -\- 4. z  =  34..      J 

5.  23 a; +  25 2/ =  915.  10.    5ic- 14^  =  11. 

6.  13 a; +  11 2/ =  414.  H.    13 a;  +  11 2;  =  103,  | 

7.  6a;+72/+4.=122,    1  7.-52/  =  4.          J 


lla;+8i/-62=145.J      12.    14  a;  -  11  y  =  29. 

13.   A  farmer  buys  horses  at  $  111  a  head,  cows  at  $  69, 
and  spends  $  2256.     How  many  of  each  does  he  buy  ? 


360  ELEMENTS  OF  ALGEBRA 

14.  A  drover  buys  slieep  at  $  4  a  head,  pigs  at  $  2,  and 
oxen  at  ^17.  If  40  animals  cost  Mm  $  301,  how  many  of 
each  kind  does  he  buy  ? 

15.  I  have  27  coins,  which  are  dollars,  half-dollars,  and 
dimes,  and  they  amount  to  $  9.80.  How  many  of  each  sort 
have  I  ? 

16.  A  drover  buys  sheep  at  $  3.50  a  head,  turkeys  at 
$  1.33i,  and  hens  at  |  0.50.  If  100  animals  cost  him  $  100, 
how  many  of  each  does  he  buy  ? 


CHAPTER   XXVIII 
THEORY  OF  LIMITS 

359.  A  variable  is  a  quantity  which  is,  or  is  conceived  to 
be,  continually  changing  in  value. 

E.g.^  the  time  since  any  past  event  is  a  variable  ;  so  also  is  the 
height  of  an  ascending  or  a  descending  balloon. 

The  amount  of  water  in  a  cistern  which  is  being  filled  by  a  con- 
tinuous stream  is  a  variable ;  and  the  number  which  measures  this 
amount  is  a  variable  number. 

Variable  numbers  are  usually  represented  by  the  final 
letters  of  the  alphabet,  as  x,  ?/,  z. 

A  constant  is  a  quantity  whose  value  is  fixed  or  invari- 
able. Constant  yiumbers  are  usually  represented  by  figures 
or  the  first  letters  of  the  alphabet,  as  4,  7,  a,  6,  c. 

E.g.^  the  time  between  any  two  past  events  is  a  constant ;  and  the 
number  which  measures  this  time  is  a  constant  number. 

360.  Limit  of  a  variable.  When,  according  to  its  law  of 
change,  a  variable  approaches  indefinitely  near,  and  con- 
tinually nearer  a  constant,  but  can  never  reach  it,  the 
variable  is  said  to  approach  the  constant  as  its  limit. 

E.g.^  let  A,  B,  and  N  be  three ^xed  points  in  the  straight  line  AN; 
then  AB  and  NB  will  be  constant  distances. 

P 

I I I I I. J 

A  O  D  E  B  N 

Suppose  a  point  P,  starting  from  A,  moves  1/2  the  distance  from 
^to  5,  or  to  C,  the  first  second  ;  1/2  the  remaining  distance,  or  to  i), 
the  next  second  ;  1/2  the  remainin<?  distance,  or  to  E,  the  next  second  ; 
and  so  on  indefinitely  ;  then  AP,  NP,  and  5P  will  be  variable  distances. 

361 


362  ELEMENTS   OF  ALGEBRA 

The  variable  distance  AP  will  approach  indefinitely  near  and  con- 
tinually nearer  the  constant  distance  AB,  but  can  never  reach  it ; 
that  is,  the  variable  distance  AP  will  approach  AB  as  its  lijnit.  The 
variable  distance  NP  will  approach  indefinitely  near  and  continually 
nearer  the  constant  distance  NB,  but  can  never  reach  it ;  that  is,  the 
variable  distance  NP  will  approach  NB^  as  its  limit.  The  variable 
distance  BP  will  approach  indefinitely  near  and  continually  nearer 
zero,  but  can  never  reach  it ;  that  is,  BP  will  approach  zero  as  its 
limit. 

The  variable  AP  will  always  be  less  than  its  limit  AB  ;  and  the 
variable  NP  will  always  be  greater  than  its  limit  NB. 

Again,  suppose  the  number  of  sides  of  a  regular  polygon  inscribed 
in  a  given  circle  is  increased  from  4  to  8  ;  from  8  to  16  ;  from  16  to  32, 
and  so  on  indefinitely  ;  then  the  area  of  the  inscribed  polygon  will 
approach  the  area  of  the  circle  as  its  limit,  and  the  area  between  the 
perimeter  of  the  polygon  and  the  circumference  of  the  circle  will 
approach  zero  as  its  limit. 

Some  variables  change  according  to  such  laws  that  they 
approach  limits,  others  do  not.  The  theory  of  limits  ap- 
plies only  to  such  variables  as  approach  limits. 

361.  Notation.  The  sign  =  is  read  ^approaches  as  a 
limit ' ;  thus  x  =  a  \&  read  ^  x  approaches  a  as  its  limit.' 

The   phrase   '  as  a  limit '   is   sometimes   omitted,   x  =  a 

being  read  'x  approaches  a.' 

The  expression  It  {x)  is  read  ^  the  limit  of  x.^ 

Thus,  It  (x)  =  a,  read  '  the  limit  of  x  is  equal  to  a,^  is 

only  another  way  of  writing  x  =  a. 

362.  The  difference  between  a  variahle  and  its  limit  ap- 
proaches zero  as  its  limit.     That  is,  if  jr  =  a,  then  jr  —  a  =  0. 

Conversely,  if  the  difference  between  a  variable  and  a  con- 
stant approaches  zero  as  its  limit,  the  variable  approaches  the 
constant  as  its  limit.     That  is,  if  jr  —  a  =  0,  then  x  =  a. 

Proof.  X  —  a  approaches  just  as  near  to  0  as  a;  does  to 
a  and  no  nearer ;  hence  if  ic  =  a,  then  x  —  a  =  0;  and 
conversely. 


THEORY  OF  LIMITS  363 

363.  A  variable  cannot  approach  two  unequal  limits  at  the 
same  time. 

Proof.  In  approaching  as  a  limit  the  more  remote  of  two 
unequal  constants,  a  variable  would  evidently  reach  a  value 
between  the  two  constants,  and  thereafter  while  it  ap- 
proached the  one  as  a  limit,  it  would  recede  from  the  other, 
which  therefore  is  not  a  limit.     Hence  the  theorem. 

364.  Tlie  limit  of  the  sum  of  a  constant  and  a  variable  is 
the  sum  of  the  constant  and  the  limit  of  the  variable. 

That  is,  if  jr  =  a,  c  4-  Jr  =  c  +  fl- 

Proof  The  sum  c -{- x  approaches  just  as  near  to  c  +  a 
as  X  does  to  a,  and  no  nearer ;  hence  if  x  =  a,  c  -{-  x  =  c-\-  a. 

365.  TJie  limit  of  the  product  of  a  constant  and  a  variable 
is  the  product  of  the  constant  and  the  limit  of  the  variable. 

That  is,  if  x  =  a,  ex  =  ca,  when  c^O. 

Proof     By  §  362,  if  a;  =  a,  a:  -  a  =  0. 

Choose  a  constant  k  as  small  as  you  please ;  then  x  —  a 
will  become  arithmetically  less  than  k/c. 

Hence  cx  —  ca  will  become  arithmetically  less  than  k. 
But  cx  —  ca  cannot  become  0,  since  x  —  a  cannot. 
Hence  when  x  =  a,  cx  =  ca. 

366.  If  two  variables  are  always  equal  and  one  approaches 
a  limit,  the  other  approaches  the  same  limit. 

That  is,  if  /  =jr  and  x  =  a,  then  /  =  a. 

Proof  li  y  =  Xj  then  y  approaches  just  as  near  to  a  as  ic 
does,  and  no  nearer ;  hence,  if  x  =  a,  y  =  a. 

367.  If  two  variables  are  always  equal  and  each  approaches 
a  limit,  their  limits  are  equal. 

That  is,  if  /  =  jr,  and  x  =  a,  and/  =  6,  then  b  =  a. 


364  ELEMENTS  OF  ALGEBRA 

Proof.     If  y  =  cc  and  x  =  a,  then,  by  §  366,  y  ^  a.         (1) 
But  by  hypothesis,  y  ==b.  (2) 

By  (1),  (2),  and  §  363,     b  =  a. 

368.  If  It  (i^)  =  0  and  It  (w)  =  0,  then  It  (vw)  =  0. 

Proof  If  ?;  =  0  and  w  =  0,  vw  approaches  nearer  to  0 
than  either  v  or  w;  but  vw  cannot  equal  0,  since  neither  v 
nor  w  can. 

Hence  if  v  =  0  and  w  ^^  0,  vw  =  0. 

369.  TJie  limit  of  the  variable  sum  of  two  or  more  variables 
is  the  sum  of  their  limits. 

That  is,  if  x  =  a,  y  =  b,  z  =  c,  •-•, 

then  x+y  +  z-\ =  ff  +  6  +  c  H . 

Proof     Let  Vi  =  x  —  a,  V2  =  y  —  b,  Vs  =  z  —  c,  -"; 
then  (aj  4-  2/  +  2  +  •••)  -  (a  +  6  +  c  +  ...)  =  '^1  +  ^2  +  ^3  +  •  -. 

Now,  however  small  a  constant  k  may  be,  each  one  of  the 
n  variables  Vi,  v^,  v^,  •••  can  become  arithmetically  less  than 
k/n ;  hence  their  sum  can  become  arithmetically  less  than  k. 
But,  since  x-\-y  -\-  z  -\-  '••  is  variable,  '^i  +  i^2  +  '^3  +  •••  cannot 
reach  and  remain  zero.     Hence  I'l  +  Vg  +  %  +  •  •  •  =0. 

Hence         x-\-y  +  z-\ =  a  +  b+c-\ .  §362 

370.  The  limit  of  the  variable  product  of  two  or  more  vari- 
ables is  the  product  of  their  limits. 

That  is,    lt(jr/z...)  =  It(jr)  •  lt(/)  •  lt(z)  .... 

Proof.     Let  x  =  a,  y  =  b, 

and  let  v  =  x  —  a,  w  =  y  —b; 

then  x  =  a-\-Vj  y  —  h  -\-u). 


THEORY  OF  LIMITS  365 

Hence      oey  =  ab  -{-  aw  -\-  bv  -{-  vw. 

.:  It  (xy)  =  a6  +  It  (aw  -^bv-{-vw)  §§  367,  364 

=  ab-{-alt(w)+blt(v)^\t(vw)    §§369,365 

=  ab  §  368 

=  lt(a^).lt(2/).  (1) 

By  (1),        It  (xy  •  zu)  =  It  (xij)  •  It  (zu) 

=  lt(x)  '\t(y)  'lt(z)  At(u). 
And  so  on  for  any  number  of  variables. 

371.  TJie  limit  of  the  variable  quotient  of  two  variables  is 
the  quotient  of  their  limits,  when  the  limit  of  the  divisor  is  not 
zero. 

That  is.  It  (jr//)  =  It  (jr)/lt  (/),  when  It  (/)  ^  0. 
Proof     Let  z  =  x/y,  or  x  —  yz-, 

then  \t(z)  =  \t{x/y),  (1) 

and  It  (x)  =  It  {yz)  =  It  (y)  •  It  (z).  (2) 

Dividing  (2)  by  It  (2^)  when  It  (y)  ^  0,  we  obtain 

It  (2)  =  It  (x)  /It  (2/),  when  It  {y)  ^  0.  (3) 

Equating  the  two  values  of  It  (2)  in  (1)  and  (3),  we  have 

\t(x/y)  =  \t{x)/\t{y),  when  \t(y)  ^  0. 
Ex.  it(^U;i^  §371 

_lt(a;).lt(y).lt(g)  „  3^^ 

c  .  It(tj)  .  It(Mj)    '  ^ 

372.  Lt  (jr")  =  [It(jr)]'',  where  n  is  a  positive  integer. 
Proof.     It  (af*)  =  It  (a;  •  ic  •  a;  •  •  •  to  w  factors)        by  notation 

=  It  (x)  '\t(x)'"ton  factors  §  370 

=  [It(aj)]".  by  notation 

E.g.,  if  X  =  a,  lt(x*)  =  a*,  lt(x6)  =  a«,  lt(a;")  =  a". 


366  ELEMENTS   OF  ALGEBRA 

Exercise  128. 
It  x  =  a,  y  =  b,  z  =  c,  u  =  e,  v  =  0,  find : 

1.  lt(aa;).  4.    lt(x^-\-a^).  7.    lt(a^/f), 

2.  \t(cx  +  av).        5.    \t(a:^-x').  8.    It  (2  a^/y'). 

3.  lt(xy-7zu).       6.    It (xy^-zu^).        9.    lt(vx'^y/z*). 
10.    It  (a.-^i/''  4-  mafz^  +  wi?).         12.    It  (a^y  +  mic^;^  +  wa?^^). 


11. 


373.  TF/ien  ^/ie  quotient  of  two  variables  or  the  product  or 
sum  of  two  or  more  variables  is  equal  to  a  constant,  the 
quotient,  product,  or  sum  of  their  limits  is  equal  to  the  same 
constant. 

Proof,    (i)  Let  xy  =  m  -,  tlien  xyz  =  mz. 

We  multiply  by  z  to  make  the  members  variable. 

.-.  It  (x)  .  It  (?/)  .  It  .  (0)  =  m  .  It  {z).  §§  370,  365 

Divide  by  It  (z),     It  (x)  •  It  {y)  =  m. 
(ii)  Let  X  ^  y  =  m  ;  then  x  =  my. 

.'.  It  (a;)  =  It  (wiy)  =  w  •  It  («/). 
.-.  It  (x)  H-  It  (y)  =  m,  when  It  {y)  =^  0. 
(iii)  Let  x  +  y  -{-  z  +  •••  =  m.  (1) 

y  +  z  +  •"  =  m  —  X. 
.'.  lt(2/)+lt(^)+...=:m-lt(a;). 
.-.  lt(x)+lt(2/)+lt(2;)...  =m. 

374.  Lt  (c/y)  =  c/lt  (y),  ivhen  It  (/)  ^  0. 
Proof.   Let  z  =  c/y;  then  zy  =  c. 

Hence  It  (0)  =  It  (c/y),  (1) 

and  It  (2!)  .  It  (y)  =  c.  §  370 

Hence  It  (z)  =  r/lt  (y),  when  It  (y)  ^  0.         (2) 

From  (1)  and  (2),         It  {c/y)  =  c/lt  (?/),  when  It  («/)  ^t  0. 


THEORY  OF  LIMITS  367 

m  m 

375.  Lt(jr'')  =  [lt(x)]''. 


Proof.   Let 

;s  =  x"  ;  then  s"  =  x^. 

(1) 

From  (1), 

lt(0)=lt(X-), 

(2) 

and 

[lt(e)]"=[lt(x)]'». 

§§  367,  372 

Hence  It  (2)=  [It  (x)]".  (3) 

m  m 

From  (2),  (3),  It  (x«)  =  [It  (x)]  « 

Ex.  It  (x^)  =  [It  (x)]^  =  J  when  x  =  a. 

376.  An  infinitesimal  is  a  variable  whose  limit  is  zero. 
Thus,  the  difference  between  a  variable  and  its  limit  is 

an  infinitesimal. 

In  approaching  its  limit  zero,  an  infinitesimal  becomes  indefinitely 
small  and  continually  smaller,  but  it  never  equals  zero.  A  small 
quantity  becomes  an  infinitesimal  when  it  begins  to  approach  zero  as 
its  limit  rather  than  when  it  reaches  any  particular  degree  of  small- 
ness.  A  quantity,  however  small,  which  does  not  approach  zero  as 
its  limit  is  not  an  infinitesimal. 

377.  An  infinite  is  a  variable  which  under  its  law  of 
change  can  exceed  any  constant  however  great. 

Thus,  the  reciprocal  of  an  infinitesimal  is  an  infinite. 

E.g.^  if  x  =  0,  1/x  can  exceed  any  constant  number  however  great ; 
thus,  since  1/(0.1")  =  10",  we  have 

when  x  =  .l,  .l^o,  .l^'^',  .V-^,  .\^^\  ..., 

1/x  =  10,  1010,  10^00,  101000,  1010000^  .... 

The  general  symbol  for  an  arithmetic  infinite  is  oo ;  and 
a;  =  00  is  read  '  x  increases  without  limit,'  ov  ^  x  is  infinite.' 
A  positive  infinite  is  denoted  by  +  oo,  and  a  negative  infinite 
by  —  00,  read  ^  00  is  a  negative  infinite.' 

An  infinite  does  not  approach  a  limit,  but  increases 
arithmetically  without  limit. 


368  ELEMENTS   OF  ALGEBRA 

378.  Any  number  which  is  neither  an  infinitesimal  nor 
an.  infinite  is  called  a  finite  number.  All  the  numbers  con- 
sidered prior  to  this  chapter  are  finite  numbers. 

379,  Any  finite  number,  not  zero,  divided  by  an  infini- 
tesimal is  an  infinite;  and  conversely,  any  finite  number, 
not  zero,  divided  by  an  infinite  is  an  infinitesimal. 

That  is,  when  x  =  0,  a/x  =  oo  (where  a  ^  0). 
And  conversely,  when  a;  =  oo,  a/x  =  0. 


Ex.    rmdltfi^i=l^i+i^   whenx  = 


4  x^  -  3  x2  +  5  _  4  -  3/x  +  6/x» 

7x-3  +  4x-8~7  +  4/x2  -  S/x^' 


173 


.    ^    /4x3-3x2  +  5\      lt(4-3/x  +  5/x3).  «« 367, 371 

V7a;3  +  4x-8J      \t(7+4/x'^-S/x^) 

=  4/7.  §§  364,  369,  379 


Exercise  129. 

Find  the   limit   of   each   of    the   following   expressions, 

when  a;  =  00  : 


1. 


3. 


Ix'-Sx  '                ^     (Sx-4:)(5x-\-A) 

5a;2  +  9*  '      9a^4-8a;-ll' 

ax^-bx-\-e  ^     (3  +  2  a;') (2  a; -  7) 

mar^  +  ca^'^  +  na;*  *    (5  a;2  + 7)  (7  +  9  a;)' 

(3ar^-l)^  g     (3  +  2a^)(a;-5) 

x'  +  9  '    (4a^-9)(l  +  ^) 


If  a;  =  a,  y  =  b,  z  =  c,  find  the  value  of : 

7.  abc,  if  xyz  —  m.  10.    Lt  (m/ar^  +  n/i/^). 

8.  ab/c,  if  xy/z=^n.  11.    Lt  (?i/a;**  + /i/y"). 

9.  ayc^,  li  xyz'^h.  12.    Lt  ((xYt" -}- ^V?/"*). 


THEORY  OF  LIMITS  369 


13.    Lt(2/^).  14.    Lt(a;V)-  15.    Lt  (V^). 


16.    Ltf^4^^V  17.    Ltf^-^^^\ 

\xu^  —  nz^J  V  2^         n^'x^J 


380.   Fractions  which  assume  the  form  0/0. 
Substituting  1  for  x  in  the  fraction  (x-  —  l)/(x  —  1),  we  have 

x2  -  1  ^  1  -  1  ^  0 
x-1      1-1     0* 

That  is,  by  the  method  in  §  12  we  fail  to  obtain  any  definite  value 
for  the  fraction  (x^  -  l)/(x  —  1),  when  x  =  1. 
But  for  any  value  of  z  other  than  1,  we  have 

(x2_i)/(x-l)  =  x  +  l.  (1) 

Hence  (1)  holds  true  when  x  =  1. 

•••""'Uffl)- ""'!(-+!)  =  2.  (2) 

That  is,  2  is  the  Umit  which  the  fraction  approaches  when  x  =  1. 
The  first  member  of  (2)  is  read  'the  limit  of  (x^  -  l)/(x-  1), 
when  X  =  1.' 

The  example  above  suggests  the  following  definition  : 

The  value  of  an  expression  for  any  particular  value  of  its 
variable  is  the  Umit  which  the  expression  approaches  when 
the  variable  approaclies  this  particular  value  as  its  limit. 

This -definition  applies  to  any  expression,  but  we  shall 
use  it  only  when  the  simpler  one  in  §  12  fails. 

Ex.   Find  the  value  of  (x^  +  x  -  2)/(x2  -  1),  when  x  =  1. 
Putting  1  for  x  in  this  fraction,  we  obtain  the  form  0/0. 
Hence  to  find  the  value  of  this  fraction,  when  x  =  1,  we  must  find 
its  limit  when  x  =  1. 

For  values  of  x  other  than  1 ,  we  have 

x'^  +  a;-2_(x-  l)(x  +  2^_x  +  2 
x2-l     ~(x-l)(x  +  l)~x  +  l' 


imit  /xg  +  x-2\      limit  /x  +  2\ 


limit 

X 


_lt(x  +  2)_3 
~lt(x+l)      2* 


370  ELEMENTS   OF  ALGEBBA 

Hence  the  value  of  the  fraction  when  aj  =  1  is  3/2. 
Note  that  we  cannot  apply  §  371  to  the  given  fraction,  for  the 
limit  of  its  divisor  is  0  when  x  =  \. 

Observe  that  the  indeterminate  form  0/0  arises  here  by 
reason  of  one  condition  ;  viz.  x  —  some  particular  value.  In 
any  such  case  an  indeterminate  form  simply  indicates  that 
the  method  of  evaluation  by  substitution  (§  12)  fails,  and 
that  the  more  general  method  by  limits  must  be  used. 

Exercise  130. 
Find  the  value  of  each  of  the  following  expressions : 

1. 


2. 


^     ^,  when  x  =  l. 
x  —  1 

^     x'-l 

'•    0^-1' 

when  x  =  l. 

^  +  J,  whenx=      1. 

.T^  —  a^ 

when  x  =  a. 

X^  —  4: 

x^  —  a^' 

when  x  =  a. 

-^+J»f2      hen.=      3. 
x'  —  9 

3     ^-f-a' 
x^-^-a^' 

when  x  =  — 

9.    (^ -«')',  when 

x  =  a. 

(x-a)i 

^     of  —  ax  —  Sx  —  3 

x'-a' 

— ,  when  X  -. 

=  —  a. 

381.  a/0,  or  absolute  infinity.  The  expression  a/0,  read  '  a  by- 
zero,'  frequently  occurs  in  mathematics,  and  the  question  arises  'what 
does  it  mean  ?  '  By  §  353,  a/0  must  symbolize  that  of  which  no  part 
can  be  expressed  by  any  number  however  large ;  hence  it  symbolizes 
that  which  transcends  all  number,  or  absolute  infinity,  of  which  we 
can  have  no  positive  idea. 

The  expression  a/0  is  commonly  denoted  by  the  symbol  co.  When 
this  notation  is  adopted,  this  meaning  of  go  must  be  clearly  distinguished 
from  that  in  §  377,  where  oo  denotes  an  infinite,  or  a  variable  which 
increases  without  limit. 

In  this  book  x  never  denotes  a/0. 


THEORY  OF  LIMITS  371 

382.  Certain  combinations  of  0  and  a/0  ;  as 

(a/0)  -  (a/0),  (a/0)  •  0,  a/0  -  a/0,  etc., 

produce  additional  indeterminate  forms.  But  any  expression  which 
assumes  any  one  of  these  forms  can  be  reduced  to  an  identical  expres- 
sion which  for  the  same  values  of  its  variables  will  assume  the  funda- 
mental form  0/0. 

E.g.,  we  have  the  identities, 

ia/x)^Ca/y)=y/x  (1) 

{a/x)  '  y  =  ay/x  (2) 

a/z-a/y  =  aiy-x)/  (xy) .  (8) 

If  X  =  0  and  y  =  0,  (1),  (2),  and  (3)  become 
(a/0)  -  (a/0)  =0/0 
(a/0)  .0  =  0/0 
a/0 -a/0  =  0/0. 

K  in  the  identity  a;«-y  =  x-/xf 

we  put  X  =  0  and  y  =  a,  we  obtain 

00  =  0/0. 

That  is,  0^  is  an  indeterminate  form. 

LAWS  OF   INCOMMENSURABLE   NUMBERS 

383.  The  laws,  already  proved  for  commensurable  num- 
bers, are,  by  the  theory  of  limits,  easily  proved  for  incom- 
mensurable numbers. 

384.  Proof  of  the  fundamental  laws. 

Let  a,  6,  c  be  any  incommensurable  constant  numbers, 
and  let  x,  y,  z  be  commensurable  variable  numbers  such  that 
X  =  a,  y  =  b,  z  =  c. 

Proof  of  (A).  x-^y  =  y  -\-x.  §  36 

.  •.  It  (x)  -h  It  (y)  =  It  (y)  +  It  (a?)  ;       §  §  367,  369 
that  is,  a  +  b  =  1  -\-a. 


372  ELEMENTS  OF  ALGEBRA 

Proof  of  (^') .                   xy  =  yx.  §  49 

•.  It  (x)  It  (y)  =  It  (y)  It  (x);  §  §  367,  370 
that  is,                               a-b  =  b  -a. 

Proof  of  (O).          {x-^y)z  =  xz  +  yz.  §  60 

.%  [It  (x)  +  It  (2/)]  It  (z)  =  It  (x)  It  (2)  +  It  (y)  It  (;2) ;  §  §  369,  370 

that  is,  (a -\- b)  c  =  ac -{-  be. 

Laws  (B),  (B'),  and  (C)  follow  from  laws  (A),  (A'),  and 
(C),  as  in  §§  36,  49,  and  88. 

385.  If  X  is  commensurable  and  jr  =  0,  then  a'  ==  1,  when 
a=^l  or  0. 

E.g.,  giving  to  x  the  successive  values  |,  J,  |,  •••,  and  finding  the 
corresponding  values  of  16*,  we  obtain  the  results  below  : 

When      x  =  l,  I,      |-,      j\,      ^V      s^f»      ih^      ^h  ■-i 
16^  =  4,  2,  1.4,  1.19,  1.09,  1.04,  1.019,  1.009  .... 

Observe  that  each  value  of  16*  is  the  square  root  of  the  preceding 
value. 

From  this  table  of  values  it  is  evident  that  when  x  =  0,  16*=  will 
approach  indefinitely  near  and  continually  nearer  1 ;  but  it  cannot 
reach  1,  since  x  cannot  reach  0,  and  16"  =  1,  when,  and  only  when, 
x  =  0. 

386.  Meaning  of  a'",  m  incommensurable. 

Let  x<m<Zf 

where  x  and  z  are  commensurable,  and  let  a>l;  then  in 
harmony  with  the  meaning  of  commensurable  exponents 
we  assume  a"*  to  denote  a  number  such  that 

a'Ka'^K  a'.  (1) 

Let  x=:m,  and  2;  =  m ; 

then  It  (z-x)^  It  (z)  -  It  (x)  =  0. 

Hence  by  §  385,  lt(a^-^)  =  1.  (2) 


THEORY  OF  LIMITS  373 

Now  a'-a'  =  a^  (a'-'  -  1). 

.-.  It  (a*  -  a')  =  It  (a')  [It  (a'-=^)  -  1]     §  §  370,  364 

=  lt(a^)(l-l)=0.  by  (2) 

From  (1),  a'  —  a' >  a""  —  a"" ', 

hence,  as  a'  —  a'  =  0,  a""  —  a'  =  0, 

,'.  a""  =  It  (a')  when  x  =  m. 

That  is,  any  base  a  icith  an  incommensurable  exponent  m 
denotes  the  limit  of  a*  when  x  =  m. 

387.   Proof  of  laws  of  exponents  I,  II,  IV,  V. 

Let  m  and  n  be  any  incommensurable  constant  numbers, 
and  let  x  and  y  be  commensurable  variable  numbers  such 
that  x^m,  y  =  n. 

Proof  of  I.                   a'a^  =  a'^".  §  345 

Hence                    It  (a'a*)  =  It  (0*+")  =  ar+".  (1) 

But                        It  (a'a^)  =  It  (a*)  •  It  (a")  =  a'^a^  (2) 

From  (1)  and  (2),      a'^a''  =  a"»+".  Law  I 

Proof  of  11.     By  law  I  we  have 

(i^~^a!*  =  a"*. 

Hence  by  §  32           ci— "  =  a'^/a\  Law  II 

Proof  of  IV.                a'b'  =  (aby.  §  348 

Hence                    It  (a'6*)  =  It  [(«6)*]  =  (a^)-  (o) 

But                        It  (a'b')  =  It  {a')  •  It  (6^)  =  a^  •  &«.  (6) 

From  (5)  and  (6),     (aft)*"  =  a-^ft"*.  Law  IV 

iVoo/  0/  V.               a'/b'  =  (a /by.  §  348 

Hence                  It  (a'^/b^)  =  It  [(«/6)^]  =  (a/ft)"*.  (7) 

But                       It  (a'/b^)  =  It  (a-)  /  It  (6^)  =  a'"/ 6"*.  (8) 

From  (7)  and  (8),  (a/b^  =  a'^/b'^.  Law  V 


374  ELEMENTS   OF  ALGEBRA 

388.  To  prove  law  III  for  incommensurable  exponents 
we  need  the  following  theorem  of  limits : 

Ify  and  z  are  commensurable  and  m  and  c  are  incommen- 
surable; tlieny  when  y  =  m  and  z  =  c,  z^  =  c"'. 

Proof.  For  all  values  of  z  (except  0  and  1),  and  hence 
for  z  variable,  by  §  386,  we  have 

2;y  _  2;'*  =  0,  when  y  =  m.  (1) 

By  §  375,  g'^  _  c"*  =  0,  when  z  =  c.  '        (2) 

From  (1)  and  (2),  (2^  -  z^)  +  {z^  -  C')  =  0 ; 

that  is,  2^  —  c"*  =  0,  when  y  =  m  and  2  =  0. 

Hence  z^  =  c"*  when  y  =  m  and  z  =  c.  (3) 

Proof  of  III.     Using  the  same  notation  as  in  §  387,  we 

have 

{ay  =  rt^^. 

Hence  It  [(a^)^]  =  It  (a^^  =  a^^.  (4) 

But  by  (3),  It  [(a^)^]  =  [It  (a^)]"^^)  =  {ar)\  (5) 

From  (4)  and  (5),         («'")''=«""*. 

From  these  laws  for  incommensurable  numbers  the  other 
laws  follow  by  the  proofs  already  given  for  commensurable 
numbers. 

VARIATION. 

389.  Two  variables  are  often  so  related  that  the  value  of 
one  depends  upon  the  value  of  the  other. 

E.g.^  the  distance  a  train  runs  at  a  given  speed  depends  upon  the 
time  it  runs,  and  this  distance  increases  when  the  time  increases. 

The  length  of  an  elastic  cord  depends  upon  its  tension,  and  this 
length  varies  when  the  tension  varies. 

liy  =  b  x'^,  the  value  of  y  depends  upon  the  value  of  x,  and  y  varies 
when  X  varies. 

We  shall  here  consider  only  the  simplest  kinds  of  variation. 


THEORY  OF  LIMITS  375 

390.  Direct  variation.  When  the  ratio  of  two  variables 
is  a  constant,  either  variable  is  said  to  vary  directly  as  the 
other. 

The  symbol  oc,  read  ^  varies  directly  as/  is  called  the  sym- 
bol of  direct  variation.  When  placed  between  two  variables 
it  denotes  that  their  ratio  is  some  constant. 

The  word  '  directly '  is  sometimes  omitted. 

E.g.,  2/ OCX,  read  ^y  varies  directly  as  a:,'  denotes  that  y/x  =  m^ 
where  m  is  some  constant. 

Again,  if  y  =  3  x,  y/x  =  3  ;  hence,  yocx  or  xccy. 

391.  If  one  vanable  varies  directly  as  another,  either  vari- 
able is  a  constant  multiple  of  the  other;  and  conversely. 

Proof     If  yc^x,y/x  =  m]   .-.  y=mx,  or  x=(l/m)ys 

Conversely,  if  y  =  mx,  y/x  =  m\   .-.yocx,  or  xccy. 

E.g.,  the  area  of  a  rectangle  =  base  into  altitude. 
Hence  if  the  altitude  is  constant,  the  area  oc  the  base. 
And  if  the  base  is  constant,  the  area  oc  the  altitude. 

392.  If  y^x,  and  if  x\  y'  and  «",  ?/"  are  any  two  sets  of 
corresponding  values  of  x  and  y,  then 

2/':a:'  =  y':a:".  (1) 

Proof     If  2/ oca:,  y'/x^  =  m  and  y'^/x^^  =  m.  (2) 

From  the  equal  ratios  in  (2),  we  have  the  proportion  (1). 
Conversely,  if  y' :  x'  =  y"  :  x",  y  =  mx  and  yocx. 

393.  Inverse  variation.  One  variable  is  said  to  ixiry  in- 
versely as  another  when  the  first  varies  as  the  reciprocal  of 
the  second. 

That  is,  /  varies  inversely  as  jr,  when  y  oc  1/jr. 

394.  If  one  variable  varies  inversely  as  another,  the  product 
of  the  two  variables  is  a  constant;  and  conversely. 

Proof.  If  yaz  1/x,  y  =  m (1/x) ;  .-.  yx  =  m. 
Conversely,  if  yx  =  m,  y  =  m(\/x);  .-.  yocl/x. 
E.g.,  if  yx  =  3,  y  varies  invereely  as  x. 


376  ELEMENTS   OF  ALGEBRA 

395.  Joint  variation.  One  variable  is  said  to  vary  as  two 
others  jointly  when  it  varies  as  the  product  of  the  two. 

That  is,  /  varies  as  x  and  z  jointly^  lohen  y  (jz  {xz),  or 

y  =  m(xz). 

E.g.,  it  W=  the  amount  of  work  done  by  2lf  men  in  Z>  days  ;  then, 
if  31  and  D  both  vary,  WazM  x  D  ; 

if  M  is  constant,  TT  x  D  ; 

if  D  is  constant,  W^M. 

One  variable  is  said  to  vary  directly  as  a  second,  and  in- 
versely as  a  third,  when  it  varies  as  the  product  of  the 
second  into  the  reciprocal  of  the  third. 

That  is,  y  varies  directly  as  x,  and  inversely  as  z,  when 

yccx(l/z),  or  y  =  mx(l/z). 
E.g.,  if  yz  =  Sx,  y  =  3x(l   z);  hence  yccx(\/z'). 

396.  In  each  of  the  preceding  cases  of  variation,  the 
value  of  the  constant,  m,  can  be  found  when  any  set  of 
corresponding  values  of  the  variables  is  known. 

Ex.  1.  Given  y<xx,  and  y  =  Q  when  x  =  2;  find  the  constant  ratio 
of  y  to  X. 

Since  y  x.x,  y  =  mx,  where  m  is  some  constant.  (1) 

Since  y  =  6,  when  oj  =  2,  from  (1)  we  have 

6  =  2  m,  orm  =  3  =  y/x. 

Ex.  2.  The  volume,  V,  of  a  pyramid  varies  jointly  as  its  height, 
H,  and  the  area  of  its  base,  B.  When  the  area  of  the  base  is  60 
square  feet  and  the  height  14  feet,  the  volume  is  280  cubic  feet.  Find 
the  area  of  the  base  of  a  pyramid  whose  volume  is  390  cubic  feet,  and 
whose  height  is  26  feet. 

Since  VccBH,  V  =  niBH,  where  m  is  some  constant.  (1) 

Substituting  the  given  values  of  V,  B,  II,  in  (1),  we  have 

280  =  m  X  60  X  14,  or  m  =  ^. 


THEORY  OF  LIMITS  877 

Hence,  when  V=  390,  and  A  =  26,  we  have 

390  =  ^  5  X  26. 
/.  B  =  45,  the  number  of  sq.  ft.  in  base. 

397.  The  simplest  way  to  treat  variations  is  to  convert 
them  into  equations. 

Ex.   If  M  Qc  y  and  yccx,  prove  that  uccx. 
Since  uccy  and  yxx,  by  §  391  we  have 

V  =  ay  and  y  =  6x,  where  a  and  6  are  some  constants. 
.'.  u  =  abx.    .'.  uccx.  §  390 

398.  If  u  ccx  when  y  is  constant j  and  u  ccy  when  x  is  conr 
stantj  then  u  ccxy  when  x  and/  both  vary. 

Let  x',  y\  vi!  be  one  set  and  a;",  2/",  w"  another  set  of  cor- 
responding values  of  a;,  y^  z,  when  all  change  together. 

Let  X  change  from  a;'  to  a;",  y  remaining  constant,  and 
suppose  that  in  consequence  u  changes  from  m'  to  u^ ;  then 
since  ucnx  when  y  is  constant,  by  §  392  we  have 

v)  :  a;'  =  u^ :  a;".  (1) 

Now  let  y  change  from  1/'  to  y",  x  remaining  constant; 
then  u  will  change  from  u^  to  w";  hence  as  Mocy  when  x 
is  constant,  we  have 

u^'.y^  =  v!^'.y\  (2) 

Multiplying  (1)  by  (2),  and  dividing  the  antecedents  by 

w,,  we  have 

v)  :  x'y'  =  u"  :  x'Y. 

Hence  uccxy.  §  392 

Similarly  it  may  be  proved  that,  if  u  varies  as  each  one 
of  the  three  variables  «,  y,  z  when  the  other  two  are  con- 
stant, then  u  oc  xyz  when  they  all  change ;  and  so  on. 

E.g.,  let  A  denote  the  area  of  a  triangle,  B  Its  base,  and  H  its 
altitude ;  then 

AccB,  when  H  is  constant, 

and  AccH,  when  B  is  constant ; 

hence  A  oc  BH,  when  B  and  H  both  change. 


378  ELEMENTS   OF  ALGEBRA 


Exercise  131. 


1 .  If  ic  varies  directly  as  y,  and  ?/  =  7  when  ic  =  18 ; 
find  X  when  y  =  21. 

2.  If  2/  varies  inversely  as  x,  and  y  =  4:  when  a;  =15; 
find  2/  when  a;  =  12. 

3.  If  aj  varies  jointly  as  y  and  2;,  and  x  =  6  when  2/ =  2 
and  z  =  2',  find  ic  when  ?y  =  5,  z  =  7. 

4.  li  x^ccy  and  2;^  x 2/,  then  0^2;  oc y. 

5.  If  icocl^2/j  ^^i<i  y  =  ^  when  a;  =  15;   find  y  when 
x  =  6. 

6.  If  a;  varies  directly  as  y  and  inversely  as   z,  and 
a;  =  10  when  y  =  15  and  z  =  6;  find  a;  when  y  =  S,  z  =  2. 

7.  If   a;  varies  directly  as   y  and  inversely  as   z,  and 
a;=14  when  2/=!^  ^.nd  2;  =  14;    find  2;  when  a; =49,  y=4:5. 

8.  If  ajocl-i-2/?  and  yccl-i-z,  prove  2; oca;. 

9.  If    3a5  H-72/<x3  a?  + 13  2/,    ^^^    y=^    when    x  =  5', 
find  the  equation  between  x  and  y, 

10.  If  the  cube  of  x  varies  as  the  square  of  y,  and  if 
a;  =  3  when  y  =  5',  find  the  equation  between  x  and  y. 

11.  If  the  area  of  a  circle  varies  as  the  square  of  its 
radius,  and  if  the  area  of  a  circle  is  154  square  feet  when 
the  radius  is  7  feet ;  find  the  area  of  a  circle  whose  radius 
is  10  feet  6  inches. 

12.  The  velocity  of  a  falling  body  varies  directly  as  the 
time  during  which  it  has  fallen  from  rest,  and  the  velocity 
at  the  end  of  2  seconds  is  64.  Find  the  velocity  at  the  end 
of  5  seconds. 

13.  The  volume  of  a  sphere  varies  directly  as  the  cube 
of  its  radius,  and  the  volume  of  a  sphere  whose  radius  is 
1  foot  is  4.188  cubic  feet.  Find  the  volume  of  a  sphere 
whose  radius  is  3  feet. 


THEORY  OF  LIMITS  379 

14.  The  pressure  of  a  gas  varies  jointly  as  its  density 
and  its  absolute  temperature;  also  when  the  density  is  1 
and  the  temperature  300,  the  pressure  is  15.  Find  the 
pressure  when  the  density  is  3  and  the  temperature  is  320. 

15.  The  volume  of  gas  varies  directly  as  the  absolute 
temperature  and  inversely  as  the  pressure.  Also  when  the 
pressure  is  15  and  the  temperature  280,  the  volume  is  1 
cubic  foot.  Find  the  volume  when  the  pressure  is  20  and 
the  temperature  300. 

16.  The  distance  through  which  a  heavy  body  will  fall 
from  rest  varies  directly  as  the  square  of  the  time,  and  a 
body  will  fall  through  144  feet  in  3  seconds.  Find  how  far 
it  will  fall  in  2  seconds. 

17.  The  pressure  of  wind  on  a  plane  surface  varies 
jointly  as  the  area  of  the  surface  and  the  square  of  the 
wind's  velocity.  The  pressure  on  a  square  foot  is  1  pound 
when  the  wind  is  moving  at  the  rate  of  15  miles  per  hour. 
Find  the  velocity  of  the  wind  when  the  pressure  on  a 
square  yard  is  16  pounds. 

18.  The  volume  of  a  right  circular  cone  varies  jointly  as 
its  height  and  the  square  of  the  radius  of  its  base ;  and  the 
volume  of  a  cone  7  feet  high  with  a  base  whose  radius  is 
3  feet  is  66  cubic  feet.  Find  the  volume  of  a  cone  9  feet 
high  with  a  base  whose  radius  is  14  feet. 


CHAPTER  XXIX 
THE  PROGRESSIONS 

399.  A  series  is  a  succession  of  terms  whose  values  are 
determined  by  some  one  law. 

A  series  is  said  to  be  finite  or  infinite  according  as  the 
number  of  its  terms  is  limited  or  unlimited. 

In  this  chapter  we  shall  consider  only  the  three  forms  of 
series  which  are  called  the  arithmetic,  the  geometric,  and  the 
harmonic  progressions. 

ARITHMETIC   PROGRESSIONS. 

400.  An  arithmetic  progression  (A.  P.)  is  a  series  in  which 
the  difference  between  any  term  (after  the  first)  and  the  pre- 
ceding term  is  the  same  throughout  the  series. 

The  difference,  which  can  be  either  positive  or  negative, 
is  called  the  common  difference. 

E.g.,  the  series 

2,     6,     8,     11,      14,      17,     20,     23,     ...,  (1) 

and  7,     5,     3,       1,  -1,  -3,   -5,  -7,     .-.,  (2) 

are  arithmetic  progressions. 

In  series  (1)  the  common  difference  is  3,  and  in  (2)  it  is  —  2. 
If  in  (2)  we  add  —  2  to  any  term,  we  obtain  the  next  term. 

401.  The  nth  term.  Let  d  denote  the  common  difference 
in  an  A.  P.,  and  a  the  first  term ;  then,  by  definition, 

the  second  terln  =  a  -\-  d, 

the  third  term    =a  +  2d, 

and  the  nth  term       =  a  +  (w  —  l)d  (1) 

380 


THE  PROGRESSIONS  381 

E.g.,  a  the  first  term  of  an  A.  P.  is  4,  and  the  common  difference 
is  5, 

the  ninth  term  =  4  +  (  9  -  1)5  =  44, 

and  the  twenty-first  term  =  4  +  (21  -  1)5  =  104. 

Ex.   The  fourth  and  fifty -fourth  terms  of  an  A.  P.  are,  respectively, 
64  and  —  61.     Find  the  twenty-seventh  term. 


I  (a 


Here  64  =  the  fourth  term  =  a  -\-  Sd<, 

and  -  61  =  the  fifty-fourth  term  =  a  +  53  d. 

Solving  system  (a),  we  find  a  =  71J,  d  =  —  5/2. 

.*.  the  twenty-seventh  term  =  a  -\-  2Qd  =  6 J. 

402.  When  three  numbers,  «,  b,  c,  are  in  A.  P.,  the  middle 
term  b  is  called  the  arithmetic  mean  of  the  other  two  terms 
a  and  c. 

403.  If  a,  6,  c  are  in  A.  P.,  by  definition  we  have 

b  —  a  =  c  —  b. 

.-.  b  =  (a-\-c)/2. 

That  is,  the  arithmetic  mean  of  any  two  numbers  is  half 
their  smn. 

404.  All  the  terms  between  any  two  terms  of  an  A.  P. 
may  be  called  the  arithmetic  means  of  the  two  terms. 

The  following  example  illustrates  how  any  number  of 
arithmetic  means  can  be  inserted  between  any  two  numbers. 

Ex.   Insert  9  arithmetic  means  between  50  and  80. 
Since  there  are  9  arithmetic  means,  80  must  be  the  eleventh  term. 
60  being  the  first ;  hence,  by  definition,  we  have 

the  eleventh  term  =  50  +  10  cZ  =  80. 

Hence  d  =  3,  and  the  required  series  is 

60,     53,     56,     59,     62,     65,     68,     71,     74,     77,     80. 


382  ELEMENTS   OF  ALGEBRA 

405.  Sum  of  n  terms.  Let  I  denote  the  nth  term,  and  S 
the  sum  of  n  terms  of  an  A.  P. ;  then 

^  =  a  +  (a  +  d)  +  (a  +  2  fZ)  H i.(i-d)-\-l, 

or  ^  =  /  4-  (Z  -  a^)  +  (Z  _  2  d)  +  ...  -f  (a  +  (^)  +  a. 

Adding  the  corresponding  terms,  we  have 

2  S  =  (a  +  I)  -\-  (a  -^  I)  -{-  (a  -\- 1)  -{-  '"  to  71  terms. 

.:  S  =  in(a  +  l).  (1) 

From  §  401,  i  =  a-\- {n -l)d.  (2) 

.-.  >S  =  inl2a-{-(n-l)dl.  (3) 

If  any  three  of  the  five  numbers  a,  d,  I,  71,  S  are  given, 
the  other  two  can  be  found  from  equations  (1)  and  (2),  or 
from  (3)  and  (2). 

Ex.  1.    Find  the  sum  of  20  terras  of  the  A.  P. 
-5-1  +  8  +  7+ 11  +  -. 
Here  a  =  -  5,  d  =  4,  n  =  20. 

.'.  S=^n{;2a+(n-  l)d} 
=  10{-10  +  19  X  4} 
=  660. 

Ex.  2.  Find  the  sum  of  the  first  n  consecutive  odd  numbers, 
1,  3,  5  .... 

Here  a  =  1,  d  =  2,  n  =  n. 

.-.  S=^n{2a-\-{n-l)d} 

=  iw{2+(>i-l)2} 

Hence  the  sum  of  n  consecutive  odd  numbers,  beginning  with  1, 
is  w2. 

Ex.  3.  The  first  term  of  an  A.  P.  is  6,  and  the  sum  of  25  terms  is 
25.     Find  the  common  difference. 


THE  PROGRESSIONS  383 

Here  a  =  6,  /S'  =  25,  w  =  25  ;  hence  from  (3)  of  §  405  we  have 
25  =  ^  X  25(12  + 24 d} 
.-.  d  =  -6/12. 

Ex.  4.  How  many  terms  must  be  taken  of  the  series  11,  12,  13, 
...  to  make  410? 

Here  a=ll,  (Z  =  l,  S  =  410  ;  hence  from  (3)  of  §  405  we  have 

410  =  in{22+(n-l)}.  (1) 

.-.  w  =  20,  or  -41. 

Since  the  number  of  terms  must  be  an  arithmetic  whole  number, 
the  number  of  terms  is  20.     See  §  297. 

Ex.  5.  How  many  terms  must  be  taken  of  the  series  —  16,  —  15, 
-14,  ...  to  make  -  100? 

Here  a  =  —  16,  d  =  1,  S  =  —  100  ;  hence  we  have 

-100  =  in{-32+(n-l)}. 

.-.  n  =  8,  or  25. 

Hence  the  number  of  terms  is  8  or  25. 

The  sum  of  the  17  terms  following  the  first  8  must  therefore  be 
zero.  These  17  terms  are  —  8,  —  7,  —  •••,  7,  8,  and  their  sum  is 
evidently  zero. 

Exercise  132. 

1.  Find  the  twenty-seventh  and  forty -first  terras  in  the 
series  5,  11,  17,  •••. 

2.  Find  the  seventeenth  and  fifty-fourth  terms  in  the 
series  10,  11^,  13,  •••. 

3.  Find  the  twentieth  and  thirteenth  terms  in  the  series 
_.S    _2   —1    ... 

4.  If  the  twelfth  term  of  an  A.  P.  is  15,  and  the  twen- 
tieth term  is  25,  what  is  the  common  difference  ? 

5.  The  seventh  term  of  an  A.  P.  is  5,  and  the  twelfth 
term  is  30.     Find  the  common  difference. 

6.  The  first  term  of  an  A.  P.  is  7,  and  its  third  term  is 
13.     Find  the  tenth  term. 


384  ELEMENTS   OF  ALGEBRA 

7.  The  first  term  of  an  A.  P.  is  20,  and  its  sixth  term  is 
10.     Find  the  twelfth  term. 

8.  The  seventh  term  of  an  A.  P.  is  5,  and  the  fifth  term 
is  7.     Find  the  twelfth  term. 

9.  Which  term  of  the  series  5,  8,  11,  •••  is  65  ? 

10.  Which  term  of  the  series  J,  f,  f ,  ♦••  is  18  ? 

11.  Insert  6  arithmetical  means  between  8  and  29. 

12.  Insert  7  arithmetical  means  between  269  and  295. 

13.  Insert  15  arithmetical  means  between  67  and  43. 

14.  If  a,  b,  c,  d  are  in  A.  P.,  prove  that  a  -\-  d  =  b  -^  c. 

15.  The  sum  of  the  second  and  fifth  terms  of  an  A.  P.  is 
32,  and  the  sum  of  the  third  and  eighth  is  48.  Find  the 
first  term. 

16.  The  sum  of  the  third  and  fourth  terms  of  an  A.  P. 
is  187,  and  the  sum  of  the  seventh  and  eighth  terms  is  147. 
Find  the  second  term. 

Find  the  sum  of  each  of  the  following  series : 

17.  5,  9,  13,  •..  to  19  terms. 

18.  1,  21,31   ...  to  12  terms. 

19.  —  5,  —  1,  3,  •••  to  20  terms. 

20.  i,  I,  i,  ...  to  7  terms. 

21.  10,  -%S  ^8-,  ...  to  7  terms. 

22.  I,  1,  |,  ...  to  15  terms. 

How  many  terms  must  be  taken  of : 

23.  The  series  42,  39,  36,  ...  to  make  315? 

24.  The  series  15,  12,  9,  ...  to  make  45  ? 

25.  The  series  —  8,  —  7,  -  6,  ...  to  make  42  ? 


TUE  PROGRESSIONS  385 

26.  Find  the  sum  of  all  the  numbers  between  100  and 
500  which  are  divisible  by  3. 

27.  Find  the  sum  of  all  the  odd  numbers  between  100 
and  200. 

28.  The  sum  of  10  terms  of  an  arithmetical  series  is  145, 
and  the  sum  of  its  fourth  and  ninth  terms  is  5  times  the 
third  term.     Determine  the  series. 

29.  Divide  80  into  4  parts  which  are  in  A.  P.,  and  which 
are  such  that  the  product  of  the  first  and  fourth  parts  is  | 
of  the  product  of  the  second  and  third. 

30.  Find  4  numbers  in  A.  P.,  such  that  the  sum  of  their 
squares  shall  be  120,  and  that  the  product  of  the  first  and 
last  shall  be  less  than  the  product  of  the  other  two  by  8. 

31.  If  a  body  falling  to  the  earth  descends  a  feet  the 
first  second,  3  a  the  second,  5  a  the  third,  and  so  on ;  (1)  how 
far  will  it  fall  during  the  tth.  second  ?  (2)  how  far  will  it 
fall  in  t  seconds  ?  Ans.  (2t  —  l)  a,  at-. 

32.  How  many  strokes  does  a  common  clock  make  in  12 
hours  ? 

33.  A  debt  can  be  discharged  in  a  year  by  paying  $  1  the 
first  week,  $  3  the  second  week,  $  5  the  third,  and  so  on. 
Find  the  last  payment  and  the  amount  of  the  debt. 

34.  One  hundred  apples  are  placed  on  the  ground  at  the 
distance  of  a  yard  from  one  another.  How  far  will  a  person 
travel,  who  shall  bring  them,  one  by  one,  to  a  basket,  placed 
at  a  distance  of  a  yard  from  the  first  apple  ? 

35.  Two  boys  A  and  B  set  out  at  the  same  time,  to  meet 
each  other,  from  two  places  343  miles  apart,  their  daily 
journeys  being  in  A.  P. ;  A's  common  difference  being  an 
increase  of  two  miles,  and  B's  a  decrease  of  5  miles.  On 
the  day  at  the  end  of  which  they  met,  each  travelled  exactly 
20  miles.     Find  the  duration  of  each  journey. 


386  ELEMENTS   OF  ALGEBRA 

GEOMETRIC   PROGRESSIONS. 

406.  A  geometric  progression  (G.  P.)  is  a  series  in  which 
the  ratio  of  any  term  (after  the  first)  to  the  preceding  term 
is  the  same  throughout  the  series. 

This  ratio,  which  can  be  either  positive  or  negative,  is 
called  the  common  ratio. 


•-g-, 

,  the  series 

2, 

6, 

18, 

54, 

162, 

8, 

4, 

2, 

1, 

h 

1, 

-1, 

h 

-i 

^, 

or 

is  a  geometric  progression  (G.  P.).     In  the  first  series,  the  common 
ratio  is  3  ;  in  the  second  series  it  is  1/2  ;  and  in  the  last  it  is  —  2/3. 

If  we  multiply  any  term  in  either  series  by  the  common  ratio,  the 
product  will  be  the  next  term  of  that  series. 

407.    The  nth.  term.     Let  r  denote  the  common  ratio,  and 
a  the  first  term  of  any  G.  P. ;  then  by  definition 

the  second  term  =  ar^ 

the  third  term     =  ar^, 

and  the  nth.  term       =  ar'*"^  (1) 

E.g.^  if  the  first  term  of  a  G.  P.  is  8,  and  the  common  ratio  is  1/2, 
the  fifth  term  =  8  x  (l/2)5-i  =  1/2, 
and  the  ninth  term  =  8  x  (l/2)9-i  =  1/32. 

Ex.    The  sixth  term  of  a  G.  P.  is  156,  and  the  eighth  term  is  7644. 
Find  the  seventh  term. 

Here  156  =  the  sixth  term  =  ar^,  (1) 

and  7644  =  eighth  term  =  ar^.  (2) 

Divide  (2)  by  (1),  49  =  r2.  (3) 

.-.  r  =  ±  7.  (4) 

But  the  seventh  term  =  sixth  term  x  r 

=  156  (±7;  =  ±1092. 


THE  PROGRESSIONS  387 

408.    Sum  of  n  terms.     Let  S  denote  the  sum  of  n  terms ; 

then 

S  =  a-{-  ar  +  a?*^  +  •••  -|-  ar**-^  +  «?•"-* 


=  a  (1 -\- r -{- 7^  +  r^  ■] h  ^'""^  +  r**-^) 

1  —  r" 
=  a- 

1  —  7* 

_a(l  — r") 


§  129 


Hence     S  =  ^^^^^ ^.  (1) 

1  —  r 

Let  Z  denote  the  ?ith  term ;  then  from  §  407 

I  =  ar'^-K  (2) 


From  (1)  and  (2) 

c,     a  —  rl 


(3) 


If  any  three  of  the  five  numbers  a,  I,  n,  r,  s,  are  known, 
the  other  two  may  be  found  from  equations  (1)  and  (2), 
or  from  (2)  and  (3). 

Ex.   Sum  the  series  6,  —  18,  64,  •••  to  6  terms. 
Here  a  =  C,  r  =  -  18  -=-  G  =  -  3,  n  =  6. 

=  f{l-36} 

=  -  1092. 

409.  When  three  numbers,  a,  b,  c,  are  in  G.  P.,  the  middle 
term  b  is  called  the  geometric  mean  of  the  other  two  terms 
a  and  c. 

410.  If  a,  6,  c  are  in  G.  P.,  by  §  406,  we  have 

c:  b  =  b:  a.     .-.  6  =  Vac. 

That  is,  the  geometric  mean  of  any  two  numbers  is  the  mean 
proportional  between  them. 


388  ELEMENTS   OF  ALGEBBA 

411.  All  the  terms  between  any  two  terms  of  a  G.  P.  may 
be  called  the  geometric  means  of  the  two  terms. 

412.  To  insert  m  geometric  means  between  a  and  b. 

Calling  a  the  first  term,  h  will  be  the  (m  +  2)  th  term ; 
hence  by  (1)  of  §  407,  we  have 

h  =  ar^+\ 


•»+!/ 


r=    ^/bTa.  (1) 

Hence  the  m  means  required  are  ar,  ar^,  •••  ar"^,  in  which 
r  has  the  value  given  in  (1). 

Ex.    Insert  6  geometric  means  between  56  and  —  7/16. 
Here  a  =  56,  6  =  -  7/16,  and  w  +  1  =  7. 

^      IQ  >      2*  X  23         2 

Hence  r  =  —  1/2,  and  the  6  means  required  are 

-  28,  14,   -  7,  7/2,   -  7/4,  7/8. 

Exercise  133. 
Find  the  last  term  in  the  following  series : 
1.   2,  4,  8,  •••  to  9  terms.      2.    2,  3,  4^,  •••  to  6  terms. 

3.  3,  -3'^,  3^  •••  to  2 7i  terms. 

4.  X,  1,  1/x,  •••  to  30  terms. 

5.  The  first  term  of  a  G.  P.  is  3,  and  the  third  term  is  4. 
Find  the  fifth  term. 

6.  The  third  term  of  a  G.  P.  is  1,  and  the  sixth  term 
is  —  1/8.     Find  the  tenth  term. 

7.  The  fourth  term  of  a  G.  P.  is  0.016,  and  the  seventh 
term  is  0.000128.     Find  the  first  term. 

8.  The  fourth  term  of  a  G.  P.  is  1/18,  and  the  seventh 
term  is  —  1/486.     Find  the  sixth  term. 

9.  Insert  3  geometric  means  between  486  and  6. 


THE  PROGBESSIONS  389 

10.  Insert  4  geometric  means  between  1/8  and  128. 

11.  Insert  5  geometric  means  between  3  and  0.000192. 

12.  Insert  4  geometric  means  between  a^b~^  and  a~^b^. 

Find  the  sum  of  the  following  series : 

13.  64,  32,  16,  •••  to  10  terms. 

14.  8.1,2.7,0.9,  ..-to?  terms. 

15.  3,  -  1,  1/3,  •••  to  6  terms. 

16.  1/2, 1/3,  2/9,  ...  to  7  terms. 

17.  -  2/5,  1/2,  -  5/8,  ...  to  6  terms. 

18.  2,  -  4,  8,  ...  to  2p  terms. 

413.  When  r  <  1  arithmetically,  the  successive  terms  of 
a  G.  P.  become  smaller  and  smaller  arithmetically,  and  the 
G.  P.  is  said  to  be  a  decreasing  progression. 

414.  TJie  limit  of  the  sum  of  an  infinite  number  of  terms  of 
a  decreasing  G.  P.  is 

Proof     Prom  (1)  of  §  412,  we  have 


1-r     1 


(1) 


Now  if  r  <  1  arithmetically,  and  the  number  of  terms, 
or  n,  is  increased  without  limit,  then 

1  —  r 
Hence  from  (1),  by  §  364,  we  obtain 

The  limit  of  the  sum  of  an  infinite  number  of  terms  of  a 
series  is  often  called  the  sum  of  the  series. 


390  ELEMENTS   OF  ALGEBRA 

E.g.,  if  a  =  2  and  r  =  1/2,  we  have  the  decreasing  G.  P., 

5,  1,  1/2,  1/4,  1/8,  1/16,  1/32,  1/64,  1/128.   ....  (1) 

The  sum  of  an  infinite  number  of  terms  of  this  series  approaches  4 
as  its  limit.  For  suppose  that  we  bisect  a  line  four  inches  long,  and 
take  away  one  of  the  parts  ;  then  bisect  the  remainder,  and  take  away- 
one  of  the  parts  ;  and  continue  this  process  without  limit.  It  is  evi- 
dent that  the  part  remaining  will  approach  zero  as  its  limit,  and  the 
sum  of  the  successive  parts  taken  away  will  approach  four  inches  as 
its  limit.  But  the  numbers  of  inches  in  the  successive  parts  taken 
away  will  be  the  terms  of  series  (1).  Hence  the  sum  of  an  infinite 
number  of  terms  of  that  series  approaches  4  as  its  limit. 

Ex.  1.   Eind  the  sum  of  the  series  1,  1/2,  1/4,  •••. 
Here  a  =  1,  r  =  1/2. 

From  (2),  it  (6')  =  ^— ^  =  2. 

Ex.  2.    Find  the  sum  of  the  series  9,  —3,  1,  •••. 

Here  a  =  9,  r  =  -  1/3. 

From  (2),  ,t  (^)  =  j_^  =  f  =6i. 

Ex.  3.   Express  0.423  as  a  common  fraction. 

0.423  =  0.4232323  ...=  —  + ^  +  H  +  .". 


10      103      105 


23 
103 

H- 

"l02J 

23 
103 

«s^ 

23 
990 

23   ,   23   ,   23   , 
^^^'  10-3  +  10-5  +  10^  + 


.-.  0.423  =  ^  +  ^'^  =  IM- 

Ex.  4.   Find  the  infinite  G.  P.  whose  sum  is  18,  and  whose  second 
term  is  —  8. 

Here  ar  =  -8,  (1) 

and  -^  =  18.  (2) 

1  —  r 

Divide  (1)  by  (2),  r(l  -  r)  =  -  4/9. 

...  ,.2  _  y  _  4/9  =  0. 


.'.  r  =  -  1/3  or  4/3. 


THE  PROGRESSIONS  391 

Only  the  value  —  1/3  is  admissible  for  r,  since  the  series  is  a 
decreasing  one. 

From(l),  a  =  -8 -(-1/3)  =  24. 

Hence  the  series  is  24,  -8,  8/3,  -8/9,  .... 


Exercise  134. 
Find  the  sum  of  each  of  the  following  series : 

1.  9,  6,  4,  ....  4.   f,  -1,  I,  .... 

2.  1    -J,  I,  ....  5.   0.9,  0.03,  0.001,  .... 

3.  h  h  yV  ••••  6.    0.8,   -0.4,  0.2,  .... 

Express  as  a  common  fraction : 
7.   0.3.       8.    0.16.       9.    0.24.       10.   0.378.       11.    0.037. 

12.  Find  the  infinite  G.  P.  whose  sum  is  4,  and  whose 
second  term  is  f. 

13.  Find  the  infinite  G.  P.  whose  sum  is  9,  and  whose 
second  term  is  —  4. 

14.  If  every  alternate  term  of  a  G.  P.  is  taken  away,  the 
remaining  terms  will  be  in  G.  P. 

15.  If  all  the  terms  of  a  G.  P.  are  multiplied  by  the  same 
number,  the  products  will  be  in  G.  P. 

16.  Show  that  the  reciprocals  of  the  terms  of  a  G.  P.  are 
in  G.  P. 

17.  By  saving  1  cent  the  first  day,  2  cents  the  second 
day,  4  cents  the  third  day,  and  so  on,  doubling  the  amount 
every  day,  how  much  would  be  saved  in  a  month  of  30 
days? 

18.  Suppose  a  body  to  move  eternally  as  follows :  20  feet 
the  first  minute,  19  feet  the  second  minute,  18 J^  feet  the 
third  minute,  and  so  on.  Find  the  limit  of  the  distance 
passed  over. 


392  ELEMENTS   OF  ALGEBRA 

19.  A  ball  falling  from  the  height  of  100  feet  rebounds 
one-fourth  the  distance,  then  falling,  it  rebounds  one-fourth 
the  distance,  and  so  on.  Find  the  distance  passed  through 
by  the  ball  before  it  comes  to  rest. 

20.  If  in  problem  31  of  exercise  132,  a  =  16 ^J^"?  ^^^  ^^ng 
will  it  be  before  the  ball  in  problem  19  comes  to  rest  ? 


To  fall  100  feet,  it  takes  Vl00TT6^,  or  10\/J^,  seconds;  to 
rebound,  or  to  fall,  25  feet,  it  takes  •\/25  -^  10^^,  or  5\/J^2_^  seconds  ; 
to  rebound,  or  to  fall,  Q\  feet,  it  takes  \/6|~n(3^,  or  |  V^^,  seconds  ; 
and  so  on. 

Hence  the  time  =  10 vTS  +  2(5  VJS  +  f  ^^  +  -) 

=  30  V^  =  1%%  V579  =  7.4805  +. 


HAEMONIC  PROGRESSIONS. 

415.   An  harmonic  progression  is  a  series  of  numbers  whose 
reciprocals  form  an  A.  P. 


E.g.^  the  series 


1,  h  h  7'   •••>  or  4,   -4, 


is  an  harmonic  progression  ;  for  the  reciprocals  of  their  terms 

1,  3,  5,  7,....,  or  \,   -\,   -f,  ... 
are  in  A.  P. 

416.  When  three  numbers  are  in  harmonic  progression 
(H.  P.),  the  middle  term  is  called  the  harmonic  mean  of  the 
other  two. 

417.  Let  H  be  the  harmonic  mean  of  a  and  h ;  then  by 
§415, 

-'  77'  7  ^^®  ^^  ^-  ^• 
a    H    h 

"  H     a      b      h' 

2       1,1  rj       2  ah 

=  --{--,  or  H 


Hah  a  +  6 


THE  PROGRESSIONS  393 

418.   If  A  and  G  denote  respectively  the  arithmetic  and 
the  geometric  mean  of  a  and  6,  then  (§§  403,  409) 


2ab 


2  a  +  6 

r.AxH  =  '^±^X^^==ab  =  0\ 
2         a  +  b 

Hence  A:G=G:H. 

That  is,  the  geometric  mean  of  any  two  numbers  is  also  the 
geometric  mean  of  their  arithmetic  and  harmonic  means. 

419.  Problems  in  H.  P.  are  generally  solved  by  inverting 
the  terms,  and  making  use  of  the  properties  of  the  resulting 
A.  P. 

Ex.  The  fifteenth  term  of  an  H.  P.  is  1/25,  and  the  twenty-third 
term  is  1/41.     Find  the  series. 

Let  a  be  the  first  term,  and  d  the  common  difEerence  of  the  corre- 
sponding A.  P.  ;  then 

26  =  the  fifteenth  term        =  a  4- 14  d, 

and  41  =  the  twenty-third  term  =  a  +  22  d. 

.-.  (Z  =  2,  a  =  -  3. 

Hence  the  A.  P.  is  —  3,  —  1,  1,  3,  5,  •••, 

and  the  H.  P.  is  -  \,  -1,  1,  \,  \,  .... 

Exercise  135. 

1.  Find  the  sixth  term  of  the  series  4,  2,  IJ,  ..•. 

2.  Find  the  eighth  term  of  the  series  IJ,  1||,  2^,  •••. 

Find  the  series  in  which 

3.  The  second  term  is  2,  and  the  thirty-first  term  is  ^\. 

4.  The  thirty-ninth  term  is  ^,  and  the  fifty-fourth  term 


394  ELEMENTS  OF  ALGEBRA 

Find  the  harmonic  mean  between 
5.    2  and  4.  6.    1  and  13.  7.    {  and  J^. 

8.  Insert  2  harmonic  means  between  4  and  12. 

9.  Insert  3  harmonic  means  between  2|-  and  12. 

10.  Insert  4  harmonic  means  between  1  and  6. 

11.  If  a,  b,  c  are  in  harmonic  progression,  prove  that 
a  —  b:b  —  c  =  a:c. 


CHAPTER   XXX 
PERMUTATIONS   AND  COMBINATIONS 

420.  Fundamental  principle.  If  one  thing  can  he  done  in  m 
ivays,  and  (after  it  has  been  done  in  any  one  of  these  ways) 
a  second  thing  can  he  done  in  n  ways;  then  the  two  things  can 
he  done  in  m  x  n  ways. 

Ex.  1.  If  there  are  11  steamers  plying  between  New  York  and 
Havana,  in  how  many  ways  can  a  man  go  from  New  York  to  Havana 
and  return  by  a  different  steamer  ? 

He  can  make  the  first  passage  in  11  ways,  with  each  of  which  he 
has  the  choice  of  10  ways  of  returning  ;  hence  he  can  make  the  two 
jourueys  in  11  x  10,  or  110,  ways. 

Ex.  2.  In  how  many  ways  can  3  prizes  be  given  to  a  class  of  10 
boys,  without  giving  more  than  one  to  the  same  boy  ? 

The  first  prize  can  be  given  in  10  ways,  with  each  of  which  the 
second  prize  can  be  given  in  9  ways  ;  hence  the  first  two  prizes  can  be 
given  in  10  X  9  ways.  With  each  of  these  ways  of  giving  the  first  two 
prizes,  the  third  prize  can  be  given  in  8  ways ;  hence  the  three  prizes 
can  be  given  in  10  x  9  x  8,  or  720,  ways. 

Proof.  After  the  first  thing  has  been  done  in  any  one  of 
the  m  ways,  the  second  thing  can  be  done  in  n  different 
ways ;  hence  there  are  n  ways  of  doing  the  two  things  for 
each  of  the  m  ways  of  doing  the  first ;  therefore  in  all  there 
are  mn  ways  of  doing  the  two  things. 

This  principle  is  readily  extended  to  the  case  in  which 
there  are  three  or  more  things,  each  of  which  can  be  done  in 
a  given  number  of  ways. 

421.  The  different  ways  in  which  r  things  can  be  taken 
from  n  things,  the  order  of  selection  or  arrangement  being 

395 


396  ELEMENTS   OF  ALGEBRA 

considered,  are  called  the  permutations  of  the  n  things  taken 
r  at  a  time. 

Thus,  two  permutations  will  be  different  unless  they  con- 
tain the  same  things  arranged  in  the  same  order. 

E.g.^  of  the  four  letters  a,  6,  c,  d,  taken  one  at  a  time,  we  have  the 
four  permutations 

a,   &,   c,   d. 

Of  these  four  letters  taken  two  at  a  time,  we  have  the  twelve  permu- 
tations 

a&,  ac^  ad,  ha,  be,  bd,  ca,  cb,  cd,  da,  db,  dc. 

If  after  each  of  these  permutations  we  place  in  turn  each  of  the 
letters  which  it  does  not  contain,  we  shall  obtain  24  permutations  of 
the  four  letters  taken  three  at  a  time. 

The  number  of  permutations  of  n  different  things  taken  r 
at  a  time  is  denoted  by  the  symbol  T^.  Thus  ^P^,  ^P^,  ^P^ 
denote  respectively  the  numbers  of  permutations  of  9  things 
taken  2,  3,  4  at  a  time. 

422.  To  find  the  number  of  permutations  of  n  dissimilar 
things  taken  r  at  a  time. 

The  number  required  is  the  same  as  the  number  of  ways 
of  filling  r  places  with  n  different  things. 

The  first  place  can  be  filled  by  any  one  of  the  n  things, 
and  after  this  has  been  filled  in*  any  one  of  these  n  ways, 
the  second  place  can  be  filled  in  {ii  —  1)  ways ;  hence  with 
n  things  two  places  can  be  filled  in  n(n  —  1)  ways ;  that  is, 

-P,  =  n(n~-1).  (1) 

After  the  first  two  places  have  been  filled  in  any  one  of 

these  n  (n  —  1)  ways,  the  third  place  can  be  filled  in  (n  —  2) 

ways ;  hence  three  places  can  be  filled  in  n(n  —  l)(n  —  2) 

ways ;  that  is, 

-P,  =  n(n-l)(n-2).  (2) 


PERMUTATIONS  AND  COMBINATIONS  397 

For  like  reason,  we  have 

^P,  =  n(n  -  l)(7i  -2)(n-  3);  (3) 

and  so  on. 

From  (1),  (2),  (3),  •••,  we  see  that  in  "P^  there  are  r  fac- 
tors, of  which  the  rth  is  n  —  r  -\-l;  hence 

"P,  =  n(n-l) (n  _  2)  ...  (?i -  r  -f  1).  (A) 

If  all  the  n  things  are  to  be  taken  at  a  time,  r  =  n,  and 
(A)  becomes 

«P„  =  w(n-l)(n-2)...3.2.1.  (B) 

423.  The  continued  product  ri  (?i  —  1)  (n  —  2)  . . .  3  .  2  . 1  is 
denoted  by  the  symbol  [n,  or  w!,  either  of  which  is  read 
*  factorial  ri/ 

Thus      [4  =  4.3.2.1;     [9  =  9. 8. 7- 6. 5- [4. 

With  this  notation  (B)  in  §  422  can  be  written 

""Pn^ln.  (B) 

That  is,  the  number  of  permutations  of  n  different  things 
taken  all  at  a  time  is  factorial  n. 

Ex.  1.   In  how  many  different  ways  can  7  boys  stand  in  a  row? 
The  number  =7P7  =  7.6.5-4.3.2.1=  5040.  by  (B) 

Ex.  2.  How  many  different  numbers  can  be  formed  with  the  figures 
1,  2,  3,  4,  5,  6,  taken  four  at  a  time  ? 

The  number  required  =  ep^  =  6  .  6  •  4  •  3  =  360.  by  (A) 

424,  If  N  denote  the  number  of  jyermutations  of  n  things 
taken  all  at  a  time,  of  which  r  things  are  alike,  s  others  alike, 
and  i  others  alike;  then 

Proof  Suppose  that  in  any  one  of  the  N  permutations 
the  r  like  things  were  replaced  by  r  dissimilar  things ;  then, 
from  this  single  permutation,  without  changing  in  it  the 


398  ELEMENTS   OF  ALGEBRA 

position  of  any  one  of  the  other  n  —  r  things,  we  could  form 
|_r  new  permutations.  Hence  from  the  N  original  permuta- 
tions we  could  obtain  N\r  permutations,  in  each  of  which 
s  things  would  be  alike  and  t  others  alike. 

Similarly,  if  the  s  like  things  were  replaced  by  s  dissimilar 
things,  the  number  of  permutations  would  be  N\r\s,  each 
having  t  things  alike. 

Finally,  if  the  t  like  things  were  replaced  by  t  dissimilar 
things,  we  should  obtain  iV^[r|s|^  permutations,  in  which  all 
the  things  would  be  dissimilar. 

But  the  number  of  permutations  of  n  dissimilar  things 
taken  all  at  a  time  is  [ri. 

Hence  iV|r[s[^  =  [n. 

\n 
Therefore  N  = 


r  sit 


Ex.  1.    How  many  different  numbers  can  be  formed  by  the  figures 

2,  2,  3,  4,  4,  4,  5,  5,  5,  5  ? 

110 
The  number  =  ,-^==—  =  12600. 


[2  [3  [4 


Exercise  136. 


1.  A  cabinet  maker  has  12  patterns  of  chairs  and  7  pat- 
terns of  tables.  In  how  many  ways  can  he  make  a  chair 
and  a  table  ?  Ans.  84. 

2.  There  are  9  candidates  for  a  classical,  8  for  a  mathe- 
matical, and  5  for  a  natural  science  scholarship.  In  how 
many  ways  can  the  scholarships  be  awarded  ? 

3.  In  how  many  ways  can  2  prizes  be  awarded  to  a  class 
of  10  boys,  if  both  prizes  may  be  given  to  the  same  boy  ? 

4.  Find  the  number  of  permutations  of  the  letters  in 
the  word  numbers.  How  many  of  these  begin  with  n  and 
end  with  s  ? 


PERMUTATIONS  AND   COMBINATIONS  399 

5.  If  no  digit  occur  more  than  once  in  the  same  number, 
how  many  different  numbers  can  be  represented  by  the  9 
digits,  taken  2  at  a  time  ?     3  at  a  time  ?     4  at  a  time  ? 

6.  How  many  changes  can  be  rung  with  5  bells  out  of 
8  ?     How  many  with  the  whole  peal  ? 

7.  How  many  changes  can  be  rung  with  6  bells,  the 
same  bell  always  being  last  ? 

8.  In  how  many  ways  can  15  books  be  arranged  on  a 
shelf,  the  places  of  2  being  fixed  ? 

9.  Given  "P4  =  12  •  "Pg ;  find  n. 

10.  Given  n  :  "Pg :  :  1 :  20 ;  find  n. 

11.  Given  "Pg :  "+2p^ :  :  5  :  12 ;  find  w. 

12.  How  many  different  arrangements  can  be  made  of 
the  letters  of  the  word  commencement  ? 

Of  the  12  letters,  2  are  c's,  3  are  m's,  3  are  e's,  and  2  are  n's ; 

112 
.-.  JVr=.  ^—   =3326400. 

[2  [3  [3  [2 

13.  Find  the  number  of  permutations  of  the  letters  of  the 
words  mammalia,  caravansera,  3fississi'p2n. 

14.  In  how  many  ways  can  17  balls  be  arranged,  if  7  of 
them  are  black,  6  red,  and  4  white  ? 

Prove  each  of  the  following  relations : 

15.  n(n  —  l)(n  —  2)"'(n  —  r -\- l)\n  —  r  =  \n. 

16.  9-8.7.6/[3  =  |9/([3|5). 

17.  n(n  —  l)(n-2)"'(n  —  r-\-  l)/\r  =  \n/(\r\n  —  r). 

18.  [5[5(6/5)=[6|4;   . •.  [5 [5  <  ^6 [4. 

19.  \a\a(a-^T)-i-a  =  \a-\-l\a  —  l},  .:  |ft|a<|a  + l|a— 1. 

20.  |a|a<|a+l|a-l<|a-l-2|a-2<|a  +  3|a-3<..». 

21.  |18  —  x\x  is  least  when  x  =  9. 


400  ELEMENTS   OF  ALGEBRA 

425.  The  different  ways  in  which  r  things  can  be  taken 
from  n  things,  without  regard  to  the  order  of  selection  or 
arrangement,  are  called  the  combinations  of  the  n  things  taken 
r  at  a  time. 

Thus,  two  combinations  will  be  different  unless  they  both 
contain  precisely  the  same  things. 

E.g.^  of  the  four  letters  a,  6,  c,  (Z,  taken  two  at  a  time,  there  are 
the  six  combinations 

a6,  ac^  ad,  be,  bd,  cd. 

Taken  three  at  a  time,  there  are  the  four  combinations 

abc,  abd,  acd,  bed. 

Taken  four  at  a  time,  there  is  one  combination  only. 

The  number  of  combinations  of  n  things  taken  r  at  a  time 
is  denoted  by  the  symbol  ""C^. 

426.  To  fiyid  the  number  of  combinations  of  n  different 
things  taken  r  at  a  time. 

Every  combination  of  r  different  things  has  |_r  permuta- 
tions ;  hence,  ""Cr  \r_  will  denote  "P^;  that  is, 

^Cr\r_=''Pr 

=  n(n  —  l){n  —  2)  '"  (n  —  r  +  1). 

Hence       „o^^«(n  -  l)(n- g-(.- r  +  1)_  ^^^ 

In  applying  this  formula,  it  is  useful  to  note  that  the 
suffix  r  in  the  symbol  "(7^  denotes  the  number  of  the  factors 
in  both  the  numerator  and  denominator  of  the  formula. 

Ex.    How  many  groups  of  4  boys  are  there  in  a  class  of  17  ? 

The  number  =  i7(74  =  iLJ^jiidi  =  2380. 
4.3.2.1 


PERMUTATIONS  AND   COMBINATIONS  401 

427.   In  (C)  of  §  426,   multiplying    the    numerator   and 
denominator  of  the  fraction  by  |  n  —  r,  we  obtain 


"Gr=r-T^--  (D) 


„(„_l)(„_2)...(n-r  +  l)l!Lz 

-  r 

\r\n  —  r 

fifi  —         L_ 

^-|r|7i-r 

) 

lituting  n  —  r  for  r  in  (D),  we  obtain 

-0    -     ^    . 

^-'-|»-r|r 

(1) 

From  (D)  and  (1),      "O,  =  "a_,.  (E) 

The  relation  in  (E)  follows  also  from  the  consideration  that  for 
each  group  of  r  things  which  is  selected,  there  is  left  a  corresponding 
group  of  n  —  r  things. 

The  relation  in  (E)  often  enables  us  to  abridge  computation. 

E.g.,  i«Ci3  =  "Ca  =  iL2<Ji  =  105. 

428.   Value  of  r  which  renders  "Cr  greatest. 

"Cv,  or  |n/(|2]  |w  —  ?•)?  is  greatest  when  [r  \n  —  r  is  least. 

[a  [a  (a  +  1)  -7-  a  =  [a  +  1  |a  —  1,  etc. ; 

.-.  \a\a<  [g  +  l  |a  — 1  <  |a  +  2  |a  — 2  <  .-•. 

Hence,  ic/ien  n  is  even,  \r  \n  —  r  is  least,  and  therefore  **Cr 
is  greatest,  when  r  =  n  —  r,  or  r  =  n/2. 

Again  [6  |6  +  1  =  \b-\-l  1 6, 

and  |6  +  l|6<|6  +  2|6-l<|6  +  3|6-2<... 

Hence  when  n  is  odd,  \r  \n  —  r  is  least,  and  therefore  **C, 
is  greatest,  when  r  =  n  —  r  ±  1,  or  r  =  (n  ±  l)/2. 

i/.gf.,   [r  |18  —  r  is  least  and  ^^Cr  is  greatest  when  r  =  9. 
Again  |r  [16  —  r  is  least  and  ^^Cr  is  greatest  when  r  =  7  or  8. 


402  ELEMENTS   OF  ALGEBRA 

Exercise  137. 

1.  How  many  combinations  can  be  made  of  9  things 
,taken  4  at  a  time  ?  taken  6  at  a  time  ?  taken  7  at  a  time  ? 

The  last  number  =  ^C7=^C2  =  36. 

2.  How  many  combinations  can  be  made  of  11  things 
taken  4  at  a  time?   taken  7  at  a  time? 

3.  Out  of  10  persons  4  are  to  be  chosen  by  lot.  In  how 
many  ways  can  this  be  done  ?  In  all  the  ways,  how  often 
would  any  one  person  be  chosen? 

4.  From  14  books  in  how  many  ways  can  a  selection  of 
5  be  made,  when  one  specified  book  is  always  included? 
when  one  specified  book  is  always  excluded  ? 

5.  On  how  many  days  might  a  person  having  15  friends 
invite  a  different  party  of  10  ?  of  12  ? 

n 

6.  Given  ^C.2  =  15,  to  find  n. 

7.  Given  «+i(74  =  9  x  "Cg,  to  find  n. 

8.  In  a  certain  district  there  are  4  representatives  to  be 
elected,  and  there  are  7  candidates.  How  many  different 
tickets  can  be  made  up? 

9.  Of  8  chemical  elements  that  will  unite  one  with 
another,  how  many  ternary  compounds  can  be  formed  ? 
How  many  binary  ? 

10.  There  are  15  points  in  a  plane,  no  3  of  which  lie  in 
the  same  straight  line.  Find  how  many  straight  lines  there 
are,  each  containing  2  of  the  points. 

11.  In  a  town  council  there  are  25  councillors  and  10 
aldermen ;  how  many  committees  can  be  formed,  each  con- 
sisting of  5  councillors  and  3  aldermen  ? 

12.  Find  the  sum  of  the  products  of  the  numbers  1,  3,  5, 
2,  taken  2  at  a  time ;  taken  3  at  a  time. 

13.  Find  the  number  of  combinations  of  55  things  taken 
50  at  a  time. 


PERMUTATIONS  AND   COMBINATIONS  403 

14.  If  2«(73:'*(72  =  44:3;  find  n. 

15.  If  ^Cu  =  "Cg;  find  n ;  find  "d, ;  find  ""C^. 

16.  In  a  library  there  are  20  Latin  and  6  Greek  books ; 
in  how  many  ways  can  a  group  of  5  consisting  of  3  Latin 
and  2  Greek  books  be  placed  on  a  shelf  ? 

17.  From  3  capitals,  5  other  consonants,  and  4  other 
vowels,  how  many  permutations  can  be  made,  each  begin- 
ning with  a  capital  and  containing  in  addition  3  consonants 
and  2  vowels  ? 

18.  If  i«a  =  ''Cr+2 ;  find  r ;  find  ^C,. 

19.  From  7  Englishmen  and  4  Americans  a  committee  of 
6  is  to  be  formed;  in  how  many  ways  can  this  be  done 
when  the  committee  contains,  (1)  exactly  2  Americans, 
(2)  at  least  2  Americans  ? 

20.  Of  7  consonants  and  4  vowels,  how  many  permutations 
can  be  made,  each  containing  3  consonants  and  2  vowels  ? 

21.  When  repetitions  are  allowed,  "P^  =  7i*",  and  "P„  =  n". 

When  repetitions  are  allowed  after  the  first  place  has  been  filled  in 
any  one  of  n  ways,  the  second  place  can  be  filled  in  n  ways ;  hence 
"Pa  =  n2,  etc. 

22.  In  how  many  ways  can  4  prizes  be  awarded  to  10 
boys,  each  boy  being  eligible  for  all  the  prizes  ? 

23.  There  are  25  points  in  space,  no  4  of  which  lie  in  the 
same  plane.  Find  how  many  planes  there  are,  each  con- 
taining 3  of  the  points. 

24.  For  what  value  of  r  is  [r  \  18  —  r  least  ?  [r  |  21  —  r ? 
|r|45-r? 

25.  For  what  value  of  r  is  ^^C,  greatest?    "C,?    ^^C,? 


CHAPTER   XXXI 
BINOMIAL  THEOREM 

429.  In  §  126  the  laws  of  exponents  and  coefficients  of 
the  binomial  theorem  were  proved  for  positive  integral 
exponents  up  to  7.  These  laws  hold  for  all  exponents, 
integral  or  fractional,  positive  or  negative. 

In  this  chapter  we  shall  prove  these  laws  for  any  positive 
integral  exponent,  and  apply  them  to  all  exponents. 

430.  From  the  distributive  law  for  multiplication,  it  fol- 
lows that  if  we  take  one  term  from  each  of  any  number  of 
binomials  and  multiply  these  terms  together,  we  shall 
obtain  a  term  of  the  continued  product  of  these  binomials ; 
and  if  we  do  this  in  every  possible  way,  we  shall  obtain  all 
the  terms  of  the  continued  product  of  these  binomials. 

E.g.,  if  we  take  a  letter  from  each  of  the  three  binomials, 

(a  +  6)(a  +  &)(a  +  6), 

and  multiply  the  three  letters  together,  we  shall  obtain  a  term  of  the 
continued  product ;  and  if  we  do  this  in  every  possible  way,  we  shall 
obtain  all  the  terms  of  this  product. 

We  can  take  the  a's  from  the  three  binomials,  and  we  can  do  this 
in  one,  and  only  one,  way  ;  hence  a^  is  a  term  of  the  product. 

We  can  take  the  b  from  one  binomial  and  the  a's  from  the  other 
two,  and  we  can  do  this  in  three  ways ;  for  the  b  can  be  taken  from 
any  one  of  the  three  binomials  ;  hence  3  a^b  is  a  term  of  the  product. 

We  can  take  the  6's  from  two  binomials  and  a  from  the  third,  and 
we  can  do  this  in  three  ways  ;  hence  3  ab^  is  a  term  of  the  product. 

Finally,  we  can  take  the  &'s  from  the  three  binomials  in  one,  and 
only  one,  way  ;  hence  h^  is  a  term  of  the  product. 

Hence       (a  +  6)  (a  +  b)  (a  -f  ?>)  =  «»  +  3  a'^b  +  3  ab^  +  &». 

404 


BINOMIAL    THEOREM  405 

431.   Binomial  theorem.     Suppose  we  have 

(a -\- b)  {a -\- b)  (a -\- b)  '  • '  to  n  factors.  (1) 

If  we  take  a  letter  from  each  of  the  ?i  binomials,  and 
multiply  these  letters  together,  we  shall  obtain  a  term  of 
the  continued  product ;  and  if  we  do  this  in  every  possible 
way,  we  shall  obtain  all  the  terms  of  this  product. 

We  can  take  the  a's  from  all  the  binomials  in  one,  and 
only  one,  way ;  hence  a'*  is  one  term  of  the  product. 

We  can  take  b  from  one  binomial  and  the  a's  from  the 
remaining  (n  —  1)  binomials,  and  we  can  do  this  in  as  many 
ways  as  one  b  can  be  taken  from  the  n  binomials,  i.e.,  n, 
or  "Ci,  ways ;  hence  "C\  •  a"~^b  is  a  term  of  the  product. 

Again,  we  can  take  the  6's  from  two  binomials,  and  the 
a's  from  the  remaining  (n  —  2)  binomials,  and  we  can  do  this 
in  as  many  ways  as  two  6's  can  be  taken  from  the  n  binomials, 
i.e.,  "(72  ways ;  hence  "C2  •  a"~-6^  is  a  term  of  the  product. 

And,  in  general,  we  can  take  the  6's  from  r  binomials 
(where  r  is  any  positive  integer  not  greater  than  w),  and  the 
a's  from  the  remaining  (n  —  r)  binomials,  and  we  can  do 
this  in  as  many  ways  as  r  6's  can  be  taken  from  the  n  bi- 
nomials, i.e.,  "(7,  ways  ;  hence  "C^  •  a"~''6''  is  the  (r  -f  l)th,  or 
general,  term  of  the  product. 

The  6's  can  be  taken  from  the  n  binomials  in  one,  and 
only  one,  way ;  hence  we  have  the  term  6",  and  this  is  what 
the  general  term  '•C^a'*~''6'"  becomes  when  r  =  n. 

Hence       (a  -\-  b)  (a  -\-  b)  (a  -\-  b)  -••  to  n  factors 

=  a"  +  ^da^'-'b  +  " C^""  -2^2  _^  ...  4-  ^Cror-'b'  +  . . .  +  6".     (2) 

If  we  substitute  for  ^'Ci,  "C2,  etc.,  their  values  as  given  in 
§  426,  we  obtain  (?i  denoting  any  positive  integer) 

(a  H-  by  =  a"  +  na'^-^b  +  '^(^~^)  a^-''b^  -f  •  •  • 

if 

_^  n(n  -  1)  (n  -  2)  ...  (n  -  r  +  1)  ^n-.^.  +  ...  +  6".     (3) 

\r_ 


406  ELEMENTS   OF  ALGEBRA 

Identity  (2)  or  (3)  is  tlie  symbolic  statement  of  the  bi- 
nomial theorem. 

The  second  member  of  either  is  called  the  expansion  of 
(a -{-by. 

Observe  that  (3)  states  in  symbols  the  laws  in  §  126,  and 
that  therefore  (3)  can  be  written  out  by  these  laws. 

Note  that  the  sum  of  the  exponents  of  a  and  b  in  any 
term  is  ii. 

Ex.   Expand  (x-^  -  i/y. 

=  (X-^y  +  4  (x-=^)3(-  2/3)  +  6  (a;-2)2(_  ^3)2  +  4  (a;-2)  (  _  y3)8 

=  x-s  -  4  x--y^  4-  6  x-^f  -  4  x'V  +  y^^. 

432.   In  the  expansion  of  (a  +  by,  the  general  term 

n(n-l)(n-2)..^(n-r  +  l)  ^„_.^.  _  ^^^  ^^  _^  ^^^^  ^^^^ 

\r 

Observe  that  there  are  r  factors  in  both  the  numerator 
and  denominator  of  the  coefficient  of  the  (r  +  l)th  term. 

By  giving  to  r  the  proper  value,  we  can  find  any  term  in 
the  expansion  of  (a  +  by. 

When  n  is  a  positive  integer,  the  coefficient  of  the  (r+l)th 
term  becomes  zero  for  any  value  of  r  greater  than  n ;  hence 
there  are  /?  + 1  tei^ms  in  the  expansion  of  (a  +  by. 

Thus,  when  r  =  n  +  1, 

n(yi-l)(n-2)  -'(n-r-\-  1)  ^  n(?i-l)(n-2)  •-(n-n)_Q^ 
[r  |_r 

Ex.   Find  the  seventh  term  of  the  expansion  of  (4  x/5  —  5/2  xy. 
Here  a  =  4 x/5,  b=-5/(2x),  w  =  9,  r  =  6. 
Substituting  these  values  in  the  formula,  we  have 

the  seventh  term  = —       ) 

1.2.3.4.5.t3V  5  /   \2x  I 

=  10500x-3. 


BINOMIAL    THEOREM  407 

433.  The  coefficients  of  the  expansion  in  (2)  of  §  431  are 

i       nri       nri       nri  nip  nfl  nfl    . 

X,       L/1,       U2,       L/3,     •••,       L/„_2,       ^n-\1       Wj 

hence  the  (r4-l)th  term  from  the  beginning  is  ''Cra^~''b''j 
and  the  (r  4-  l)th  term  from  the  end  is  "C^.^a'^^""''. 

But  "O,  =  "(7„_,  for  all  values  of  r  (§  427). 

Hence,  in  the  expansion  of  (a  +  by,  the  coefficients  of  any 
two  terms  equidistant  from  the  beginning  and  the  end  are  the 
same,  so  that  the  coefficients  of  the  last  half  of  the  expansion 
can  be  written  from  those  of  the  first  half 

434.  If,  in  identity  (3)  of  §  431,  we  put  a  =  1  and  b  =  x, 
we  have 

(l+a;)-  =  l  +  nx  +  ''^V^'^^+-+.-7^^— ^"+-+^" 

2  \r\n  —  r 

This  is  a  convenient  form  of  the  binomial  theorem^  and 
one  which  is  often  used. 

Observe  that  this  form  includes  all  cases ;  e.gr.,  if  we  want  to  find 
(a  4-  ?>)",  we  have 

435.  In  (1)  of  §  434  the  coefficients  of  x,  x^,  x^,  •••,  x'^  are 
the  values  of  "Ci,  "Cg,  "C3,  •••,  "(7„;  hence  (t)  can  be  written 

(1  +  a;)"  =  1  +  "C,x  +  ^C^  +  ...  +  "Oa'  +  •••  +  ^'C^a;".     (1) 
Putting  x  =  l,  we  obtain 

2«  =  1+  "Ci  +  '^C^  +  -  +  "C,  +  -  +  "a.  (2) 

That  is,  the  sum  of  the  coefficients  in  the  expansion  of 
(1  +  xy,  or  (a  +  by,  is  2\ 

From  (2)  it  follows  also  that  the  sn7n  of  all  the  combinor 
tions  that  can  be  made  of  n  things,  taken  1,  2,  •",  n  at  a  time, 
is  2"  - 1. 


■408  ELEMENTS   OF  ALGEBRA 

Exercise  138. 

By  the  laws  in  §  126  write  the  expansion  of : 

1.  {3x'-2y)\      Q     (r-^_^h)\  n.    {x-'^-2c^)\ 

2.  (2a2-3  6y.      7.    {f-^n^)\         12.    (1  - 1  A)«. 

3.  (c^  +  hy.  8.    (2  x/^  -  3/xy.  13.    (^2/  _  «-f)5. 

4.  (3a^  +  2/)'-         9.    (x-^-afy.         14.    (2  a;/3  -  a/c)«. 

5.  (2-3i«y.       10.    (a-'--x-y.        15.    (a;-^-2?/-^)^ 

16.  Find  the  3d  term  in  the  expansion  of  (a  —  3  by^. 

17.  Find  the  7th  term  in  the  expansion  of  (1  —  xy^. 

18.  Find  the  middle  term  in  the  expansion  of  (1  +  xy^. 

19.  Find  the  middle  term  in  the  expansion  of  (2  a?  —  3  yy. 

20.  Find  the  18th  term  in  the  expansion  of  (1  +  x)^. 

21.  Find  the  7th  term  in  the  expansion  of 

[4:x/5-5/{2x)J. 

22.  Find  the  17th  term  in  the  expansion  of  (xF  —  1/x)^. 

436.    Binomial  theorem,  exponent  fractional  or  negative. 

When  the  exponent  of  a  binomial  is  fractional  or  nega- 
tive, the  laws  in  §  126,  or,  what  is  the  same  thing,  the 
formula 

(a  +  by  =  a"  +  na-~^b  +  <^lz1}  a^-^^b'  +  - 

_^n(n  -l)(n-  2)  ...  (n  -  r  +  1)  ^._,^.  ^  _^     ^^^ 

\l 

gives  an  infinite  series ;  for  in  this  case  no  one  of  the  factors 
n,  n  —  1,  n~  2,  etc.,  in  the  (r  +  l)th  term  can  ever  be  zero. 

When,  however,  r  increases  ivithout  limit,  the  sum  of  r  terms 
of  this  series  will  approach  (a  -\-  by  as  its  limit,  provided  the 
first  term  of  the  binomial  is  arithmetically  greater  than  the 


BINOMIAL    THEOREM  409 

second  term.  That  is,  when  7i  is  fractional  or  negative,  the 
infinite  series  in  (1)  is  the  expansion  of  (a  +  by  provided 
a>b  arithmetically. 

A  proof  of  this  theorem  is  too  difficult  to  be  given  here. 
For  a  rigorous  proof,  see  Taylor's  "  Calculus,"  §  98. 

Ex.  1.   Expand  (c-i  —  (P)~^  and  find  the  general  term. 
Applying  the  laws  in  §  126  we  obtain 

(c-i  _  cf^)-f  =  [(c-i)  +  (-  d2)]-t 

-i¥.(0"^'(-^^)«  +  -  (1) 

=  c^  +  f  cV  +  ^  c'^V .+  ^^  c^(P  +  -.  '         (2) 

The  two  distinct  steps,  that  of  applying  the  laws  to  obtain  (1)  and 
that  of  performing  the  indicated  operations  in  (1)  to  obtain  (2),  must 
be  taken  separately. 

In  performing  the  operations  indicated  in  (1),  first  note  the  number 
of  negative  numeral  factors  in  a  term  to  determine  the  quality  of  its 
numeral  coefficient.  Thus  in  the  fourth  term  there  are  four  negative 
factors,  —  ^^\  and  (—  l)^. 

Substituting  in  the  general  term  for  n,  a,  and  b  their  values  —  |, 
c-i,  and  —  ^2^  ^e  obtain 

the  (r+l)th  term  =  (~i)^~^)(~Y^  "•  ^~^~^"^^^(c-0"^""(-<f^/ 

[r 

~  5'"[r  ^  ^ 

Since  there  are  r  factors  in  the  numerator  in  (3),  the  term  involves 
the  2  rth  power  of  —  1,  which  is  +  1. 

In  (2),  by  this  article  d^  must  be  arithmetically  less  than  c-i. 

Ex.  2.   Expand  1/(1  +  x)  and  find  the  general  term. 
Applying  the  laws  in  §  120,  we  obtain 

(1  +  x)-i  =  1-1  -  1  .  1-2  .  x  +  1  . 1-3  a;2  -  1  .  1-*  x^-{-  "' 

=  1  -x-\-x^-x^-hx* .  (1) 

The  (r  +  l)th  term  =  (- ^^^-^  "'(- '^^  l-i-'-x'-=(-  lyxr.     (2) 
In  (1),  by  this  article  x  is  limited  to  values  between  —  1  and  +  1. 


410  ELEMENTS  OF  ALGEBRA 

Ex.  3.    (1  +  a;)-2=  1-2  -  2  . 1-3  .  x  +  3  . 1-4  .  a;2  -  4  . 1-5x3  +  ... 
=  l-2x  +  3x2-4x3  +  .... 

The  (r+l)th  tevm  =  ^~^^^~^^"'^~''~'^^l-'-'xr={-iyir-\-l)xr. 


Ex.  4.    Expand  1/ Vl  -  x,  or  (1  -  x)  2. 

(1  _  x)-^=r^  -  i  .  r*(-  X)+^-  H(-  X)2  _  ^%  .  l-^(-  X)3+  ... 
=l+lX+|x2+3:V:«^+•••. 

The  (r  +  l)th  tern  ^(-i)(-t)(-g- (-^-^'+l)i-i-(-xy 
_1.3.5...(2r-l)^, 

Ex.  5,   Find  the  cube  root  of  127. 

127  =  125  4-  2  =  53  +  2. 
.-.   v/l27=(53  +  2)i 

=  OT^  +  K5')~^2  -  i(53)"^'  22  +  ^(53)-t23  +  ... 

3     52      9     55^81     58 

=  5  +  0.0266666  -  0.0001422  +  0.0000012 

=  5.0265255  -. 

The  smaller  the  ratio  of  the  second  term  of  the  binomial  to  the  first, 
the  more  rapidly  the  successive  terms  of  the  expansion  decrease,  and 
therefore  the  fewer  the  terms  it  is  necessary  to  find. 

Here  we  put  127  =  125  +  2,  because  125  is  the  perfect  cube  which 
makes  the  ratio  of  the  second  term  to  the  first  the  smallest. 

Exercise  139. 
Expand  to  four  terms  : 

1.  (l-x)-\           5.    {l-\-x)-\  9.  (l-5a;)l 

2.  {l-x)-\           6.    (l  +  2a;)-^  10.  (6^  _  c-*)"^. 

3.  (l-x)-\           7.    (2-x)-^  11.  a/-Vx^-f. 

4.  (l-x)-\           8.    (l-3a;)"^.  12.  b/(J-b-^). 


BINOMIAL   THEOREM  411 

Find  the  general  term  in  the  expansion  of : 

13.  (l-x)-'.  15.    (1-xyK  17.    (l-2x)-^. 

14.  (l-x)-\          16.    (l  +  x)~i  18.    (l4-3a;)~i 

In  its  expansion  find  the  : 

19.  Sixth  term  and  eleventh  term  of  (i  a  —  bVb)^. 

20.  Fifth  term  of  (1  -  a^)"! 

21.  Seventh  term  of  {x~^—y^y. 

22.  Third  term  and  eleventh  term  of  (1  -f  2a;)^^. 

23.  Fifth  term  of  (c-2+ e-^)-'». 

24.  Sixth  term  of  (a;"^-ci26f)-f. 

Find  to  four  places  of  decimals  the  value  of : 

25.  ^31.      27.    ^29.  29.    </620.        31.    ^/998. 

26.  </l7.      28.    ^122.        30.    ^31.  32.    ^3128. 

Expand  to  four  terms : 

33.  (8 +  12  a)*.  38.  (9  +  2»)*. 

34.  (l-Sx)^.  39.  (4a- 8a;)"^. 

35.  (1-3  a;)-*.  40.  (c^a"*  -  6 V^)"^. 

36.  (a^  +  c^)*.  ^^  a 

37.  (c-d^l  '  (cb-^-^y'¥ 

42.    Find  the  general  term  in  each  of  the  examples  from 
33  to  39  inclusive. 


CHAPTER   XXXII 
LOGARITHMS 

437.  The  exponent  which  the  base  a  must  have  in  order 
to  equal  the  number  N  is  called  the  logarithm  of  N  to  the 
base  a. 

That  is,  if  a^  =  N,  (1) 

X  is  the  logarithm  of  N  to  the  base  a,  which  is  written 

X^IOQaN.  (2) 

Equations  (1)  and  (2)  are  equivalent;  (2)  is  the  logarith- 
mic form  of  writing  the  relation  between  a,  x,  and  N, 
given  in  (1). 

U.g.i  since  3^  =  9,  2  is  the  logarithm  of  9  to  the  base  3,  or  2  =  logs  9. 
Since  2*  =  16,  4  is  the  logarithm  of  16  to  the  base  2,  or  4  =  log2  16. 
Since  2-3  =  1/8,  -  3  =  logs  (1/8). 

Since  4^  =  8,  3/2  =  log4  8. 

Eeview  §§  52,  336,  338,  339,  386  on  exponents. 

Exercise  140. 

1.  Express  each  of  the  following  relations  in  the  loga- 
rithmic form : 

2^  =  8,  3^  =  81,  4^^  =  64,  12^  =  144,  6^  =  216,  n'  =  b. 

2.  Express  each  of  the  following  relations  in  the  expo- 
nential form : 

logs  125  =  3,  log2  32  =  5,  log4  64  =  3,  log3  81  =  4,  \og,M=b. 

412 


LOGARITHMS  413 

3.  When  fclie  base  is  3,  what  are  the  logarithms  of  1,  3,  9, 
27,  81,  243,  729  ? 

4.  When  the  base  is  4,  what  are  the  logarithms  of  1,  4, 
16,  64,  256,  1024  ? 

5.  When  the  base  is  2,  what  are  the  logarithms  of  1, 1/2, 
1/4,  1/8,  1/16,  1/32,  1/64,  1/128,  1/256  ? 

6.  When  the  base  is  10,  what  are  the  logarithms  of  1, 10, 
100,  1000,  10000,  100000,  0.1,  0.01,  0.001,  0.0001,  0.00001  ? 

7.  When  the  base  is  3,  and  the  logarithms  are  0,  1,  2,  3, 
4,  —  1,  —  2,  —  3,  —  4,  what  are  the  numbers  ? 

438.  The  logarithms  of  all  arithmetic  numbers  to  any  given 
base  constitute  a  system  of  logarithms. 

Since  1*  =  1,  1  cannot  be  the  base  of  a  system  of  log- 
arithms. Any  arithmetic  number  except  1  can  evidently 
be  taken  as  the  base  of  a  system  of  logarithms. 

Since  logarithms  are  exponents,  from  the  general  proper- 
ties of  exponents,  we  obtain  the  general 

PROPERTIES  OF  LOGARITHMS  TO  ANY  BASE. 

439.  The  logarithm  of  1  is  zero. 

Proof.  a«  =  l,         .-.  log«l  =  0. 

440.  The  logarithm  of  the  hOjSe  itself  is  1. 
Proof  a^  =  a,        .-.  \o^^a  =  l. 

441.  The  logarithm  of  a  product  is  equal  to  the  sum  of  the 
logarithms  of  its  factors. 

Proof  Let  M=a%  N=a^', 

then  MxN=a'+«.  §345 

Hence  log«  (M  x  N)  =  x-\-y  =  log„  M  -f  log^  N. 

E.g. ,  log4  (16  X  64)  =  log4  16  +  log4  64  =  2  +  3  =  5. 


414  ELEMENTS   OF  ALGEBRA 

442.  The  logarithm  of  a  quotient  is  equal  to  the  logarithm 
of  the  dividend  minus  the  logarithm  of  the  divisor. 

Proof.     Let  M=a%  N=aV', 

then  M-i-]Sr=  a^  ^  §  346 

Hence     log«  (M  -i-  N)  =  x  —  y  =  log^  M  —  log„  N. 

E.g.,  logs  (243  -  27)  =  logs  243  -  logs  27 

=  5-3  =  2. 

443.  The  logarithm  of  any  power  of  a  number  is  equal  to 
the  logarithm  of  the  number  multiplied  by  the  ex2)onent  of  the 
power. 

Proof     Let  M=a'\ 

then,  for  all  real  values  of  p,  we  have 

Jf  p  =  aP\  §  348 

Hence  log„(  Jf^)  =px=p  log„  M. 

E.g.,  log4 (163)  =  3  .  log4 16  =  3  X  2  =  6  ; 

log8(81^)=|log381  =  f  x4  =  3; 
and  logs  (25"^)  =  -  f  logs  25=-fx2=-3. 

444.  By  §  443,  the  logarithm  of  any  positive  integral 
power  of  a  number  is  equal  to  the  logarithm  of  the  number 
multiplied  by  the  exponent  of  the  power ;  and  the  logarithm 
of  any  root  of  a  number  is  equal  to  the  logarithm  of  the 
number  divided  by  the  index  of  the  root. 

Ex.  1.    Given 

logio  2  =  .30103  and  logio  3  =  .47712  ;  find  logio  ^720. 
logio  ^720  =  ^  logio  (23  X  32  X  10)  §  443 

=  K'3  logio  2  +  2  logio  3  +  logio  10)  §§  441,  443 

=  K-90309  +  .95424  +  1)  =  .95244. 


LOGARITHMS  415 

Ex.  2.     log„  [  V^  -  (63c?)  ]  =  loga  x^  -  loga  {b^c^)  §  442 

=  loga  x^  -  (log«  63  +  log^  c^)  §  441 

=  I  log«a;  -  3  loga  &  -  f  loga  c.        §  443 

445.    If  a  series  of  numbers  are  in  geometric  progression^ 
their  logarithms  are  in  arithmetic  progression. 

E.g.,  if  N=\/21,     1/9,     1/3,     1,     3,     9,     27,-. 

log3iV=-3,        -2,     -1,     0,     1,     2,       3,  .... 

Proof     The  logarithms  of  the  terms  of  the  Gr.  P. 

Ny  .JSTr,  ...,      JVr", 

are      log^iV,       log« .V  +  log„  r,  ••.,       log„  JV^  +  »i  loga r, 

which  is  an  A.  P.  whose  common  difference  is  log^  r. 

Exercise  141. 

Express  log^  y  in  terms  of  log^  b,  log„  c,  log,  ic,  and  log,  z, 
having  given  the  following  equations : 

1.  y  =  T'b^<^.  4.    y=W^- 

2.  2/=^2'-Vc«.  5.    2/=V^-VW. 


3.    y  =  '^.  6.   ,  =  ^. 

C^  ^^^& 

Given  logjo  2  =  .3010,  log^  3  =  .4771,  find : 

7.  Iogio4;   logio5;   logioO;   logioS;   logio9;   logwlO. 

logio  5  =  logio  10  -  logio  2  =  1-  .3010  =  .6990. 

8.  Iog,ol2.  10.    logio  30.  12.    logio  (3/2). 

9.  logio  16.  .11.    logio  50.  13.    logio  (6/5). 

14.    logioV^OO.  15.    logio  ^120. 

16.  Between  what  integral  numbers  does  logio -^  lie? 
when  N  lies  between  10  and  100?  Between  1  and  10? 
Between  .1  and  1  ?  Between  .01  and  .1  ?  Between  .001 
and  .01? 


416  ELEMENTS   OF  ALGEBRA 

446.    If  the  base  of  logarithms  is  greater  than  1, 

(i)  The  logarithm  of  a  number  is  positive  or  negative, 
according  as  the  number  is  greater  or  less  than  1. 

(ii)  The  logarithm  of  an  infinite  is  infinite  ;  and  the 
logarithm  of  an  infinitesimal  is  a  negative  infinite,  or,  as  it 
is  often  stated,  the  logarithm  of  zero  is  negative  infijiity. 

Proof   By  §  437  x  is  the  logarithm  of  a""  to  the  base  a. 

Let  a  >  1 ;  then,  by  the  principles  of  exponents,  we  know 
that 

if  a*  >  1,  x>0;  il  a''  <1,  x<0;  hence  (i). 

If     a*  =  00,  a;  =  00 ;  if  a"  =  0,  x  =  —  :/:>;  hence  (ii). 


COMMON   LOGARITHMS. 

447.  The  logarithms  used  for  abridging  arithmetic  compu- 
tations are  those  to  the  base  10 ;  for  this  reason  logarithms 
to  the  base  10  are  called  common  logarithms. 

Thus  the  common  logarithm  of  a  number  answers  the 
question,  ^  What  power  of  10  is  the  number  ? ' 

Most  numbers  are  incommensurable  powers  of  10;  hence 
most  common  logarithms  are  incommensurable  numbers, 
whose  approximate  values  we  express  decimally. 

Hereafter  in  this  chapter  when  no  base  is  written,  the 
base  10  is  to  be  understood. 

When  a  logarithm  is  negative,  for  convenience  it  is  ex- 
pressed as  a  negative  integer  plus  a  positive  decimal. 

E.g.,  the  conmion  logarithm  of  any  number 

between  10  and  100  is  +1  +  a  positive  fraction  ; 
between  1  and  10  is  0  +  a  positive  fraction ; 
between  0.1  and  1  is  -1  +  a  positive  fraction; 
between  0.01  and   0.1    is  -2  4-  a  positive  fraction. 


LOGARITHMS  417 

448.  The  integral  part  of  a  logarithm  is  called  the 
characteristic,  and  the  positive  decimal  part  the  mantissa. 

A  negative  characteristic  is  usually  written  in  the  form 

I,  or  9  -  10;  2,  or  8  -  10;  3,  or  7  -  10;  etc. 

E.g.,  log  434.1  =  2.63759  ;  +2  is  the  characteristic  and  .+63759  is  the 
mantissa  :  log  0.0769  =  2.88593,  or  8.88593  -  10  ;  2,  or  8  -  10,  is  the 
negative  characteristic,  and  .+88593  is  the  mantissa.  The  sign  -  is 
written  over  the  2  to  show  that  it  affects  the  characteristic  alone. 

449.  The  characteristic  of  the  common  logarithm  of  any 
number  is  found  by  the  following  simple  rule: 

Wlien  the  number  is  greater  than  1,  the  characteristic  is 
positive  and  arithmetically  one  less  than  the  number  of  digits 
to  the  left  of  the  decimal  point  ;  when  the  number  is  less  than 
1,  the  characteristic  is  negative  and  arithmetically  one  greater 
than  the  number  of  zeros  between  the  decimal  point  and  the 
first  significant  figure. 

E.g.,  785  lies  between  IO2  and  10'  ; 
hence  log  785  =  2  +  a  mantissa. 

Again  0.0078  lies  between  lO-^  and  IO-2  ; 
hence  log  0.0078  =  —  3  +  a  mantissa. 

Proof.   Let  N  denote  a  number  which  has  m  digits  to  the 
left  of  the  decimal  point ;  then  N  lies  between  10*""^  and 
10"*; 
that  is,  N  =  10<'"-^) + *  f™^"°°. 

.-.  log  -^=  (m  —  1)  +  a  mantissa. 

Again  let. ^denote  a  decimal  which  has  m  zeros  between 
the  decimal  point  and  the  first  significant  figure;  then  iV 
lies  between  lO-^""^^)  and  lO""* ; 

that  is,  N=  lO-^'^+i)  +  *  <^'=«°". 

.-.  log  N=—  (m  -h  1)  -f  a  mantissa. 


418  ELEMENTS   OF  ALGEBBA 

450.  The  common  logarithms  of  numbers  which  differ  only 
in  the  positio7i  of  the  decimal  point  have  the  same  mantissa. 

Proof.  When  a  change  is  made  in  the  position  of  the 
decimal  point,  the  number  is  multiplied  or  divided  by  some 
integral  power  of  10 ;  that  is,  an  integer  is  added  to,  or 
subtracted  from,  the  logarithm,  and  therefore  its  mantissa 
is  not  changed. 

E.g.,  log  1054.3  =  3.02296, 

log       1.0543      =  0.02296, 
log         .010543  =  8.02296  -  10,  or  2.02296. 

451.  When  a  negative  logarithm  is  to  be  divided  by  a 
number,  and  its  negative  characteristic  is  not  exactly  divis- 
ible by  that  number,  the  logarithm  must  be  so  modified  in 
form  that  the  negative  integral  part  will  be  exactly  divisible 
by  the  number. 

Ex.    Given  log  0.0785  =  2.8949;  find  log  \/0. 0785. 

Log  v/0.0785  =  I  log  0.0785  =  }  (2.8949) 

=  }(7.  +6.8949)=  1.8421. 

Adding  —5  +  5  to  the  logarithm  does  not  change  its  value  and 
makes  its  negative  part  divisible  by  7. 

Exercise  142. 

1.  Log  427.32  =  2.6307.     Find  log  42732,  log  42.732. 

2.  Log  23.95  =  1.3793.     Find  log  23950,  log  239.5, 
log  239500,  log  0.002395,  log  0.0002395,  log  2395. 

3.  Log  4398  =  3.64326.  Find  log  V0.4398,  log -^0.4398, 
log  ^439.8,  log  ^0.04398,  log  ^0.004398. 

4.  Log  674.8  =  2.82918.  Find  log  ^0.6748,  log  ^0.6748, 
log  -^0.06748,  log  ■{/0.06748,  log  ^0.006748. 

452.  Tables  of  logarithms.  Common  logarithms  have  two 
great  practical  advantages :    (i)  Characteristics  are  known 


LOGABITHMS  419 

by  §  449,  so  that  only  mantissas  are  tabulated ;  (ii)  mantis- 
sas are  determined  by  the  sequence  of  digits  (§  450),  so  that 
the  mantissas  of  integral  numbers  only  are  tabulated. 

At  the  close  of  this  chapter  will  be  found  a  table  which 
contains  the  mantissas  of  the  common  logarithms  of  all 
numbers  from  1  to  999  correct  to  four  decimal  places. 

Note.  Tables  are  published  which  give  the  logarithms  of  all  num- 
bers from  1  to  99999  calculated  to  seven  places  of  decimals ;  these  are 
called  '  seven-place '  logarithms.  For  many  purposes,  however,  the 
four-place  or  five-place  logarithms  are  sufficiently  accurate. 

From  a  table  of  logarithms  we  can  obtain : 

(i)   The  logarithm  of  a  given  number ; 

(ii)  The  number  corresponding  to  a  given  logarithm. 

453.    To  find  the  logarithm  of  a  given  number. 
Ex.  1.   Find  log  7.85. 

By  §  450,  the  required  mantissa  is  the  mantissa  of  log  785. 
Look  in  column  headed  "N"  for  78.     Passing  along  this  line  to 
the  column  headed  5,  we  find  .8949,  the  required  mantissa. 
Prefixing  the  characteristic,  we  have 

log  7.85  =  0.8949. 

Ex.  2.    Find  log  4273. 2. 

When  the  number  contains  more  than  three  significant  figures,  we 
must  use  the  principle  that  when  the  difference  of  two  numbers  is 
small  compared  with  either  of  them,  the  difference  of  the  numbers 
is  approximately  proportional  to  the  difference  of  their  logarithms. 

By  §  450,  the  required  mantissa  is  that  of  log  427.32. 

The  mantissa  of  log  427  =  .6304. 
The  mantissa  of  log  428  =  .6314. 

That  is,  an  increase  of  1  in  the  number  causes  an  increase  of  .0010 
in  the  mantissa ;  hence  an  increase  of  .32  in  the  number  will  cause  an 
increase  of  .32  of  approximately  .0010,  or  .0003,  in  the  mantissa. 

Adding  ,0003  to  the  mantissa  of  log  427,  and  prefixing  the  character- 
istic, we  have 

log4273.2  =  3.6307. 


420  ELEMENTS   OF  ALGEBBA 

Ex.3.    Find  log  0.0006049. 

By  §  450,  the  required  mantissa  is  that  of  log  604.9. 
The  mantissa  of  log  604  =  .7810. 

Also,  an  increase  of  1  in  the  number  causes  an  increase  of  .0008  in 
the  mantissa;  hence  .9  of  .0008,  or  .0007,  must  be  added  to  .7810. 

.-.  log 0.0006049  =  4.7817,  or  6.7817  -  10. 

To  find  log  30  or  log  3,  find  mantissa  of  log  300. 

Exercise  143. 
Find,  from  the  table,  the  logarithm  of  the  numbers : 

1.  8.  5.    703.  9.    0.05307.         13.    7.4803. 

2.  50.  6.    7.89.  10.    78542.  14.    2063.4. 

3.  6.3.  7.   0.178.        11.    0.50438.         15.    0.0087741. 

4.  374.         8.   3.476.         12.    0.00716.         16.    0.017423. 

454.    To  find  a  number  when  its  logarithm  is  given. 

Ex.  1.    Find  the  number  of  which  the  logarithm  is  3.8954. 
Look  in  the  table  for  the  mantissa  .8954.     It  is  found  in  line  78  and 
in  column  6  ;  hence 

.8954  =  the  mantissa  of  log  786. 
.-.  3.8954  =  log  7860; 
or  7860  is  the  number  whose  logarithm  is  3.8964. 

Ex.  2.    Find  the  number  of  which  the  logarithm  is  1.6290. 
Look  in  the  table  for  the  mantissa  .6290.     It  cannot  be  found  ;  but 
the  next  less  mantissa  is  .6284,  and  the  next  greater  is  .6294. 

Also,  .6284  =  mantissa  of  log  425, 

and  .6294  =  mantissa  of  log  426. 

That  is,  an  increase  of  .0010  in  the  mantissa  causes  an  increase  of  1 
in  the  number ;  hence  an  increase  of  .0006  in  the  mantissa  will  cause 
an  increase  of  approximately  3%  of  1,  or  .6,  in  the  number;  hence 

I  .6290  =  the  mantissa  of  log  425.6  ; 

.-.  1.6290  =  log  42.56. 


LOGARITHMS  421 

Ex.  3.   Find  the  number  of  which  the  logarithm  is  3.8418, 
Look  in  the  table  for  the  mantissa  .8418.     It  cannot  be  found  ;  but 
the  next  less  mantissa  is  .8414,  and 

.8414  =  mantissa  of  log  694. 

Also,  an  increase  of  .0006  in  the  mantissa  causes  an  increase  of  1  in 
the  number  ;  hence  an  increase  of  .0004  in  the  mantissa  will  cause  an 
increase  of  f  of  1,  or  .66  in  the  number ;  hence 

.8418  =  the  mantissa  of  log  694.66. 

.-.  3.8418  =  log  0.0069466. 

Exercise  144. 
Find  the  number  of  which  the  logarithm  is : 

1.  1.8797.                 6.   8.1648-10.  11.  3.7425. 

2.  7.6284-10.         7.   9.3178-10.  12.  7.1342-10. 

3.  0.2165.                  8.   1.6482.  13.  3.7045. 

4.  2.7364.                  9.   8.5209-10.  14.  8.7982-10. 

5.  4.0095.                10.   3.8016.  15.  3.4793. 

455.   The  cologarithm  of  a  number  is  the  logarithm  of  its 
reciprocal. 
That  is,       colog N=\og(l -^N)  =  —  log N. 

To  make  the  fractional  part  of  the  cologarithm  positive,  if 
log  iV  >  0  and  <  10,  colog  N  is  written 

(lO-logiV)-lO; 
if  log  ^  >  10  and  <  20,  colog  N  is  written 
(20-logiV^)-20. 
E.g.,  cologO.0674  =-(2.7589)=  1.2411; 

colog  432  =  (10  -    2.6263)  -  10  =  7.3737  -  10 ; 
colog  345000000000  =  (20  -  11.5378)  -  20  =  8.4622  -  20. 

Instead  of  subtracting  the  logarithm  of  a  divisor,  we  can, 
by  §  87,  add  its  cologarithm. 


422  ELEMENTS   OF  ALGEBRA 

Ex.1.     Find  the  value  of  1M^AM723. 
0.0534  X  7.238 

log   15.08  =  1.1784 
log0.0723  =  8.8591 -10 
colog0.0534  =  1.2725 
colog   7.238  =  9.1404-10 
Add,         log  (fraction)  =  0.4504  =  log  2.8213. 

Hence         the  fraction  =  2.8213. 

Ex.  2.     Find  the  value  of  0.0543  x  6.34  x  (-  5.178). 

Iog0.0543  =  8.7348 -10 
log     6.34  =  0.8021 
loa:  5.178  =  0.7141 


Add,         log  (product)  =  0. 2510  =  log  1 .  7824. 

Hence  the  product  is  —  1.7824. 

By  logarithms  we  obtain  simply  the  arithmetic  value  of  the  result ; 
its  quality  must  be  determined  by  the  laws  of  quality. 

J   5.42  X  427.2 
Ex.  3.     Find  the  value  of  Ji 


\3.244  X  0.0231^ 

log5.42  =  0.7340  =0.7340 

2  log  427.  =  (2.6304)  x  2  =  5.2608 

4  colog  3.24  =  (9.4895  -  10)  x  4  =  7.9580  -  10 
^  colog  0.0231  =  (1.6364)  -2  =  0.8182 

5)4.7710 
.-.  log  (root)  =  0.9542 

.-.  root  =  9.00 

456.   An  exponential  equation  is  one  in  which  the  unknown 
appears  in  an  exponent ;  as  2*  =  5,  af  =  10. 

Such  equations  are  solved  by  the  aid  of  logarithms, 

Ex.  1.     Solve  32*  -  14  x  3^  +  45  =  0.  (1) 

Factor  (1),  (3*  -  9)  (3^  -  5)  =  0.  (2) 


LOGARITHMS  42^ 


(4) 


Equation  (2) 

is  equivalent  to  the  two  equations 

3'  =  9,              (3)               3-  =  5. 

From  (3), 

x  =  2. 

From  (4), 

a;  log  3  =  log  5. 

log  5^  0.6990  _g49 
log  3     0.4771 

Hence  the  roots  of  (1)  are  2  and  1.4649. 

Exercise  145. 
Find  by  logarithms  the  value  of : 

1.  742.8x0.02374.  7.  4743-5-327.4. 

2.  0.3527x0.00572.  g  9.345^  (_  0.0765). 

3.  78.42x0.000437. 

_  2.476  x(- 0.742) 

4.  5234  X  (-0.03671).  9.  ^3  ^^  ^  (_o.ooi21)* 

5.  3.246  X  (-0.0746).  ^^  321  x(- 48.1)  x  (357) 

6.  -  4.278  X  (-  0.357).  *  421  x  (-  741)  x  (4.21) 

11.  5l  14.  («)«.  17.  (iiyi 

12.  0.02li  15.    714.2i  18.    (3|)^27 

13.  0.5328.                 16.    (Iff)^  19.    4.71=^ 
/0.035'*  X  54.2  X  785^  x  0.0742 


206 


20. 


21. 


22. 


10.035'^ 
^         4.72^  X  7.14^  X  8.47^ 

3/     0.0427^  X  5.27  x  0.875^ 
V7.42I*  X  Vl-'^4  X  V0.00215 

5/0.714^x0.1371^x0.0718^ 
^'  0.5242x0.742^  x  0.0527^ 


424  ELEMENTS  OF  ALGEBRA 

Solve  each  of  the  following  equations : 

23.  31^  =  23.  25.    5^  =  800.  27.   5^-3  =  8^*+^ 

24.  0.3^  =  0.8.  26.    12^  =  3528.         28.    aH^''  =  (^, 
29.   23*52^-1  =  4^^3^+1.  30.   42=^-15  (4^) +56  =  0. 

COMPOUND  INTEREST  AND   ANNUITIES. 

457.  To  find  the  compound  interest,  $  /,  and  amount,  $  M, 
of  a  given  principal,  $  P,  in  n  years,  $  r  being  the  interest  on 
$  1  for  1  year. 

Let  ^R  =  the  amount  of  $  1  in  1  year ;  then  R  =  l-\-r, 
and  the  amount  of  f  P  at  the  end  of  the  first  year  is  $  PR ; 
and  since  this  is  the  principal  for  the  second  year,  the 
amount  at  the  end  of  the  second  year  is  $  PR  x  R,  or 
$  PR^.  For  like  reason  the  amount  at  the  end  of  the  third 
year  is  $PR^,  and  so  on;  hence  the  amount  in  n  years  is 
^PR""',  that  is, 

M=PR'',  or  P(l  +  r)^  (1) 

Hence  I=P(R--1).  (2) 

If  the  interest  is  payable  semi-annually,  the  amount  of 
$P  in  1/2  a  year  will  be  $P(l-\-r/2);  hence,  as  n  years 
equals  2  n  half-years, 

M=P(l  +  r/2y\  (3) 

Similarly,  if  the  interest  is  payable  quarterly, 

M=P{l+r/4:y\  (4) 

Ex.  Find  the  time  in  which  a  sum  of  money  will  double  itself  at 
ten  per  cent  compound  interest,  interest  to  be  "converted  into  prin- 
cipal" semi-annually. 

Here  1  +  r/2  =  1.05.    Let  P  =  1 ;  then  M=2. 
Substituting  these  values  in  (3),  we  obtain 
2  =  (1.05)2« 
.-.  log2  =  2w  .log  1.05. 


2  log 


^^gl-  =  ^^^  =  7.1  years,  ^ns. 
02 1.05     0.0424  ^ 


LOGARITHMS  425 

458.  Present  value  and  discount.  Let  f  P  denote  the  pres- 
ent value  of  the  sum  $M  due  in  n  years,  at  the  rate  ?-;  then 
evidently  in  n  years  at  the  rate  r,  $P  will  amount  to  $M-j 
hence 

M=PB^,  or  P  =  MR-\ 

Let  $  D  be  the  discount ;  then 

D  =  M-  P=  3/(1  -  Z2-**). 

459.  An  annuity  is  a  fixed  sum  of  money  that  is  payable 
once  a  year,  or  at  more  frequent  regular  intervals,  under 
certain  stated  conditions.  An  Annuity  Certain  is  one  pay- 
able for  a  fixed  number  of  years.  A  Life  Annuity  is  one 
payable  during  the  lifetime  of  a  person.  A  Perpetual  An- 
nuity, or  Pei'petuity,  is  one  that  is  to  continue  forever,  as, 
for  instance,  the  rent  of  a  freehold  estate. 

460.  To  find  the  amount  of  an  annuity  left  unpaid  for  a 
given  number  of  years,  allowing  compound  interest. 

Let  ^  J.  be  the  annuity,  n  the  number  of  years,  $  R  the 
amount  of  one  dollar  in  one  year,  $  JHf  the  required  amount. 
Then  evidently  the  number  of  dollars  due  at  the  end  of  the 

First  year     =  A ; 

Second  year  =  AR  +  A ; 

Third  year    =  AR' +  AR -{- A -, 

nth  year       =  AR'-^  +  AR"  ---\ [-AR-\-A 

^A(R^-l) 
R-1 

That  is,  M=-(R''-1).  •         (1) 

Ex.  1.  Find  the  amount  of  an  annuity  of  1 100  in  20  years,  allow- 
ing  compound  interest  at  4 J  per  cent. 

r  ^  0.046 


426  ELEMENTS   OF  ALGEBRA 

By  logarithms,  1.04520  =  2.4117. 

...  3/ ^l^lill^  3137.11. 
0.045 

Hence  the  amount  of  the  annuity  is  $3137.11. 

Ex.  2.  What  sum  nmst  be  set  aside  annually  that  it  may  amount 
to  $  50,000  in  10  years  at  6  per  cent  compound  interest  ? 

rrom(l),      .4  =  -^!^=.^M00Aa06^  3793.37. 

Hence  the  required  sum  is  $3793.37. 

461.  To  find  the  preseiit  value  of  an  annuity  of  ^A  pay- 
able at  the  end  of  each  of  n  successive  years. 

Let  $P  denote  the  present  value;  then  the  amount  of 
^P  in  n  years  will  equal  the  amount  of  the  annuity  in  the 
same  time  :  that  is, 

PR^  =  A{R'-l)r-\  (1) 

.-.  P=A{l-R--)r-\  (2) 

If  the  annuity  is  perpetual,  then  n  =  oo,  i2~"  =  0,  and 

(2)  becomes 

P  =  Ar-\ 


Exercise  146. 

1.  Write  out  the  logarithmic  equations  for  finding  each 
of  the  four  numbers,  M,  R,  P,  n. 

2.  In  what  time,  at  5  per  cent  compound  interest,  will 
$100  amount  to  f  1000? 

3.  Find  the  time  in  which  a  sum  will  double  itself  at 
4  per  cent  compound  interest. 

4.  Find  in  how  many  years  f  1000  will  become  f  2500 
at  10  per  cent  compound  interest. 

5.  Find  the  present  value  of  $10,000  due  8  years  hence 
at  5  per  cent  compound  interest. 


LOGARITHMS  427 

6.  Find  the  amount  of  $1  at  5  per  cent  compound  in- 
terest in  a  century. 

7.  Show  that  money  will  increase  more  than  thirteen- 
thousand-fold  in  a  century  at  10  per  cent  compound  interest. 

8.  If  A  leaves  B  $1000  a  year  to  accumulate  for  3 
years  at  4  per  cent  compound  interest,  find  what  amount  B 
should  receive. 

9.  Find  the  present  value  of  the  legacy  in  example  8. 

10.  Find  the  present  value,  at  5  per  cent,  of  an  estate  of 
$  1000  a  year  to  be  entered  on  immediately. 

11.  A  freehold  estate  worth  f  120  a  year  is  sold  for 
$  4000  ;  find  the  rate  of  interest. 

12.  A  man  has  a  capital  of  $20,000,  for  which  he  re- 
ceives interest  at  5  per  cent;  if  he  spends  $1800  every 
year,  show  that  he  will  be  ruined  before  the  end  of  the 
17th  year. 


428 


ELEMENTS   OF  ALGEBRA 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0263 

0294 

0334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1969 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2465 

2480 

2604 

2529 

18 

2563 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2766 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3679 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4266 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4466 

28 

4472 

4487 

4502 

4518 

4633 

4548 

4664 

4679 

4694 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4767 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

6011 

6024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

6119 

6132 

6145 

6169 

5172 

33 

5185 

5198 

5211 

5224 

5237 

6250 

5263 

5276 

6289 

6302 

34 

5315 

5328 

5340 

5353 

6366 

5378 

5391 

6403 

6416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5602 

6514 

6627 

5639 

5561 

36 

5563 

5575 

5587 

5599 

5611 

6623 

6636 

6647 

6658 

6670 

37 

5682 

5694 

5705 

5717 

6729 

5740 

5752 

5763 

5776 

5786 

38 

5798 

5809 

5821 

5832 

6843 

6865 

6866 

5877 

6888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

6977 

5988 

6999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6086 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6376 

6385 

6395 

6406 

6416 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6671 

6580 

6590 

6699 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6876 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7060 

7069 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7136 

7143 

7162 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7236 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372  7380 

7388 

7396 

TABLE  OF  MANTISSAS 


429 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7436 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7613 

7520 

7528 

7536 

7543 

7661 

67 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7762 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7826 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8036 

8041 

8048 

8056 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8166 

8162 

8169 

8176 

8182 

8189 

66 

8196 

8202 

8209 

8216 

8222 

8228 

8236 

8241 

8248 

8264 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8388 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8446 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8637 

8643 

8549 

8566 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8616 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8667 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8746 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8786 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8826 

8831 

8837 

8842 

8848 

886^4 

8869 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8964 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9176 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9316 

9320 

9326 

9330 

9335 

9340 

86 

9345 

9360 

9355 

9360 

9365 

9370 

9375 

9380 

9386 

9390 

87 

9395 

9400 

9405 

9410 

9416 

9420 

9426 

9430 

9436 

9440 

88 

9445 

9450 

9465 

9460 

9466 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9613 

9518 

9523 

9528 

9633 

9638 

90 

9542 

9547 

9552 

9567 

9562 

9566 

9571 

9576 

9681 

9686 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9676 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9764 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9806 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9846 

9860 

9864 

9869 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

CHAPTER   XXXIII 


GRAPHIC  SOLUTION  OF  EQUATIONS  AND  SYSTEMS 


M' 


4--4- 


M 


P 


462.  Let  XX'  and  YY'  be  any  two  fixed  straight  lines 
at  right  angles  to  each  other  at  0.  Let  the  directions 
OX  and  OF  be  positive  directions;  then  the  directions 
^Y  .  OX'  and  0Y>  will 

be    negative    direc- 
tions. 

The  lines  XX 
and  YY'  are  called 
axes  of  reference, 
__  and  their  intersec- 
tion 0,  the  origin. 
From  P,  any 
point  in  the  plane 
of  the  axes,  draw 
PM     parallel     to 


IP 


M 


Fig.  1 


YY' ;  then  the  po- 


sition of  P  will  be 
determined     when 

we  know  both  the  lengths  and  the  directions  of  the  lines 

OMsindMP. 
The  line   OM,  or  its  numerical   measure,  is  called  the 

abscissa  of  the  point  P;  and  MP,  or  its  numerical  measure, 

is  called  the   ordinate   of  P.      The  abscissa  and  ordinate 

together  are  called  the  coordinates  of  P. 

E.g.,  OM'  and  M'P'  are  the  coordinates  of  P' ;  the  abscissa,  OW, 
is  negative,  and  the  ordinate,  M'P',  is  positive.  OM'",  the  abscissa 
of  P"',  is  positive,  and  M'"P"',  its  ordinate,  is  negative. 

430 


SOLUTION  OF  EQUATIONS  AND   SYSTEMS      431 

An  abscissa  is  usually  deaoted  by  the  letter  x,  and  an 
ordinate  by  y. 

Observe  that  the  numerical  measure  of  OM  or  MP  is  a 
positive  number,  if  it  extends  in  the  direction  OX  or  OF; 
and  a  negative  number  if  it  extends  in  the  direction  OX' 

or  or. 

The  axis  XX'  is  called  the  axis  of  abscissas,  or  the  jr-axis ; 
and  YY\  the  axis  of  ordinates,  or  the /-axis. 

The  point  whose  coordinates  are  x  and  y  is  denoted  by 

{^y  y)' 

E.g.,  (2,  —  3)  denotes  the  point  of  which  the  abscissa  is  2,  and  the 
ordinate  —  3. 

We  use  a  system  of  coordinates  analogous  to  that  explained  above 
whenever  we  locate  a  city  by  giving  its  latitude  and  longitude ;  the 
equator  ie  one  axis,  and  the  assumed  meridian  the  other. 

Ex.   Plot  the  point  (-  2,  3)  ;  (-3,-4). 

In  the  figure  lay  off  OJf  =  -  2,  and  on  MT*  parallel  to  YT  lay  off 
JiPP'  =  +  3  ;  then  P'  is  the  point  (-2,  3). 

To  plot  (-3,  -  4),  lay  off  OM"  =  -  3,  and  on  M"P"  parallel  to 
YY'  lay  ofE  M"P"  =  -  4  ;  then  P"  is  the  point  (-3,  -  4). 

The  lines  XX'  and  YY  divide  the  plane  into  four  equal 
parts  called  quadrants,  which  are  numbered  as  follows : 
XOY  is  the  Jirst  quadrant j  YOX'  the  secondy  X'OY  the 
third,  and  YOX  the  fouHh. 

Exercise  147. 

1.  Plot  the  point  (2,  3);  (4,  7);  (3,  -5);  (-2,  +3); 
(-3,  +5);  (4,  -2);  (-2,  -3);  (-5,  -3);  (-2,  4); 
(-4,-1);  (0,0). 

2.  In  which  quadrant  is  (+a,  +b)?  (+a,  "6)?  ('a,  +b)? 
(-a,-b)? 

3.  What  is  the  quality  of  x  and  of  y,  when  the  point 
(Xy  y)  is  in  the  first  quadrant  ?  Second  quadrant  ?  Third 
quadrant?     Fourth  quadrant? 


432  ELEMENTS   OF  ALGEBRA 

4.  In  which  quadrants  can  the  point  (x,  y)  be,  when  x  is 
positive  ?   X  negative  ?    y  positive  ?    y  negative  ? 

5.  In  what  line  is  the  point  (a;,  0)  ?   (0,  y)  ? 

6.  Where  is  the  point  (0,  0)  ?  (4,  0)  ?  (-  3,  0)  ?  (0,  2)  ? 
(0,-5)? 

463.   Graphic  solution  of  equations  in  x  and  /. 

The  locus,  or  graph,  of  an  equation  in  a;  and  y  is  the  line 
or  lines  which  include  all  the  points,  and  only  those,  whose 
coordinates  satisfy  the  equation. 

Ex.  1.   Draw  the  locus  of  y  =  x"^  —  x  —  6.  (1) 

If  in  (1)  we  put  a;  =  -  3,  -  2,  -  1,  ...,  we  obtain 

when  X  =  -  3,   -  2,  - 1,       0,     1/2,       1,       2,  3,  4,  ..., 

y  =  6,       0,  -  4,   -  6,  -  6|,   -  6,  -  4,  0,  6,  .... 

Drawing  the  axes  XX'  and  TY'  in  fig.  2,  and  assuming  01  as  the 
linear  unit,  we  plot  the  points 

(-3,6),  (-2,  0),  (-1,  -4),  (0,  -6),  .... 

The  relative  positions  of  these  points  indicate  the  form  of  a  curve 
through  them. 

Whenever  there  is  any  doubt  about  the  form  of  this  curve  between 
any  two  plotted  points,  as  between  (0,  —6)  and  (1,  —6),  one  or 
more  intermediate  points  should  be  found  and  plotted. 

As  X  increases  indefinitely  from  3,  y  (^or  x^  —  x  —  6)  continues  posi- 
tive and  increases  indefinitely  ;  hence  the  locus  has  an  infinite  branch 
in  the  first  quadrant.  As  x  decreases  indefinitely  from  —  2,  y  con- 
tinues positive  and  increases  indefinitely ;  hence  the  locus  has  an 
infinite  branch  in  the  second  quadrant. 

Drawing  a  smooth  curve  through  the  plotted  points  we  obtain  the 
curve  ABC  in  fig.  2,  which,  with  its  infinite  branches,  is  the  locus 
of  equation  (1). 

This  curve  is  called  the  locus  of  the  equation  because  each  and 
every  real  solution  of  equation  (1)  is  the  coordinates  of  some  point 
on  the  curve. 


SOLUTION  OF  EQUATIONS  AND   SYSTE3IS      433 

Imaginary  or  complex  solutions  of  an  equation  cannot  be 
represented  by  the  coordinates  of  any  points  in  the  plane 
XO  Y,  since  by  definition  the  coordinates  of  every  point  in 
this  plane  are  real. 

Note.  The  pupil  should  use  coordinate  or  cross-section  paper,  and 
with  a  hard  pencil  draw  the  loci  of  equations  neatly  and  accurately. 


X     X- 


t—^ 


Ex.  2.   Draw  the  locus  ot  y  =  3?  —  2x.  (1) 

When    x  =  -2,  -y^,  -1,  -0.8,  0,       0.8,       1,  y/2,  2,  ..., 
y=-4,  0,       1,       1.1,  0,  -1.1,  -1,      0,  4,  .... 

As  X  increases  indefinitely  from  2,  y  (or  x^  —  2x)  continues  positive 
and  increases  indefinitely  ;  hence  the  locus  has  an  infinite  branch  in 
the  first  quadrant. 

As  X  decreases  indefinitely  from  —2,y  continues  negative  and 
arithmetically  increases  without  limit ;  hence  there  is  an  infinite 
branch  in  the  third  quadrant. 

Plotting  these  points,  as  in  fig.  3,  and  tracing  a  smooth  curve 
through  them,  we  obtain  the  curve  ABCD,  which,  with  its  infinite 
branches,  is  the  locus  of  (1). 


434 


ELEMENTS   OF  ALGEBRA 


Ex.  3.    Draw  the  locus  oi  y  =  x^  —  S  x"^  +  i. 
When        x  =  -3/2,   -  1,  -  ^,  0,  1/2,  1,  2,  3,  ..., 
y=-  6.1,       0,    3.1,  4,    3.4,  2,  0,  4,  .... 

As  X  increases  from  2,  y  increases  indefinitely  from  0 ;  and,  as 
X  decreases  from  —!,'</  decreases  indefinitely  from  0. 

The  locus  is  the  curve  ABCD  in  fig.  4,  which  has  one  infinite 
branch  in  the  first  quadrant  and  another  in  the  third. 


2/  =  a:*  +  x3  -  3  x2  -  ic  +  2. 
A  \Y  E 


Fig.  5 


Ex.  4.   Draw  the  locus  of  y  =  x*  +  a;^  -  3  x^  —  a:  +  2. 
When     a;  =  -|,   -2,      _  |,   -  1,   -  ^,  -  .2,  0,      J,  1,      f,  ..-, 
y  =  9.2,       0,   -1.6,       0,     1.7,  2.08,  2,  0.9,  0,  2.2,  ••.. 

The  locus  is  the  curve  in  fig.  5,  which  has  one  infinite  branch  in 
the  first  quadrant  and  another  in  the  second. 

The  foregoing  examples  illustrate  how  each  real  solution  of  an 
equation  in  x  and  y  is  the  coordinates  of  some  point  in  the  locus ; 
hence,  by  the  coordinates  of  its  points,  the  locus  of  an  equation  in 
X  and  y  giveg  all  its  infinite  number  of  real  solutions. 


SOLUTION  OF  EQUATIONS  AND   SYSTEMS      435 

Exercise  148. 

Draw  the  locus  of  each  of  the  following  equations,  stating 
in  which  quadrants  the  infinite  branches  lie : 

2.  y  =  x  —  a^  +  6.  5.    y  =  x^  —  a^  —  6x. 

3.  y  =  oc^  —  4:X.  G.    y  =  x^  — 5x^  +  4:. 

7.  Using  one  set  of  axes,  draw  the  loci  of 

y  =  x,  y  =  x  +  2,  y  =  x-2. 
Observe  that  these  loci  are  parallel  straight  lines. 

8.  Using  one  set  of  axes,  draw  the  loci  of 

y  =  2x,  y  =  2x  +  3,  y  =  2x-3. 

9.  Draw  the  locus  of 

y  =  3x-2,oiy  =  -2x-\-l,of2y  =  4:X-6. 

These  examples  illustrate  the  truth  that  the  locus  of  any  linear 
equation  in  x  and  y  is  a  straight  line. 

Hence  to  draw  the  locus  of  any  linear  equation  we  can  plot  two  of 
its  points  and  draw  a  straight  line  through  them. 

10.  Draw  the  locus  of 

y  =  4,  of  2/  =  —  3,  of  X  =  3,  of  a;  =  —  4. 
Observe  that  each  of  these  loci  is  parallel  to  one  of  the  axes. 

11.  From  the  origin  0  as  a  centre  and  with  a  radius  5, 
draw  the  circumference  of  a  circle.  Draw  the  ordinate  PM 
of  any  point  P  on  this  circumference  and  the  radius  OP. 
Denote  the  coordinates  of  P  hy  x  and  y.  Then  from  the 
right-angled  triangle  OMP,  we  obtain  a?^  +  ^^  _  52^ 

What  then  is  the  locus  of  a^  -f  2/^  =  2^  ? 


436  ELEMENTS   OF  ALGEBRA 

12.  Draw  the  locus  of  or  -|-  3/-  =  9,  of  x-  -\-  y-  =  16. 

Examples  11  and  12  illustrate  the  truth  that  the  lucus  of  any  equa- 
tion of  the  form  x^  -\-  y^  =  r^  is  the  circumference  of  a  circle  whose 
centre  is  at  the  origin  and  whose  radius  is  r. 

13.  Draw  the  locus  of     4tx^  +  ^y'  =  36.  (1) 

Here  y  =±  |V9  -  xK 

Evidently  —  3  is  the  least  value  of  x  which  will  render  y  real ;  hence 
no  part  of  the  locus  can  lie  to  the  left  of  the  line  a;  =  —  3.  For  like 
reason  no  part  of  the  locus  can  lie  to  the  right  of  the  line  x  =  3- 

When         a;  =-3,   -2.5,      -2,      -1,       0,  1,  2,3, 

2/  =  0,    ±1.1,   ±1.5,    ±1.9,   ±2,   ±1.9,   ±1.5,0. 

The  locus  is  the  ellipse  RASB  (fig.  8,  page  439),  the  semi-axes 
being  3  and  2. 

Observe  that  in  (1)  the  coefficients  of  x^  and  ?/2  are  unequal,  while 
in  examples  11  and  12  they  are  equal. 

14.  Draw  the  locus  of  a;-  -f  4?/-  =  4. 

15.  Draw  the  locus  of  a.-^  —  4  ?/^  =  4. 

Here  y  =±\/2V^^~^^.  (1) 

When  X  > 2  or  <—  2,  the  values  of  y  in  (1)  are  real ;  when  x  lies 
between  —  2  and  +2,  y  is  imaginary  ;  hence  there  is  an  infinite 
branch  in  each  of  the  four  quadrants,  but  no  point  of  the  locus  lies 
between  the  lines  x  =  —  2  and  x  =  2. 

16.  Draw  the  locus  of  2/^  =  4  x. 

17.  Using  one  set  of  axes,  draw  the  loci  of 

X  —  y  =  —  3  and  a;  -f  1/  =  1. 

Observe  that  these  equations  are  independent  and  consistent,  and 
that  the  one  and  only  point  common  to  their  loci  is  (—  1,  2). 

18.  Using  one  set  of  axes,  draw  the  loci  of 

2 ic  —  2/  =  —  1  and  2x  —  y  =  —  3. 

Observe  that  these  equations  are  inconsistent,  and  that  their  loci 
are  parallel  and  hence  have  no  point  in  common. 


SOLUTION  OF  EQUATIONS  AND   SYSTEMS      437 


19.  Using  one  set  of  axes,  draw  the  loci  of 

2 a;  +  2/  =  1  and  6x-\-3y  =  3. 

Observe  that  these  equations  are  equivalent,  and  that  their  loci 
coincide  and  hence  have  all  points  in  common. 

20.  What  is  the  greatest  number  of  points  in  which  a 
straight  line  can  cut  the  locus  in  fig.  2  ?  In  fig.  3  ?  In 
fig.  4  ?  In  fig.  5  ?  Compare  each  answer  with  the  degree 
of  the  equation  of  each  locus. 

464.   Graphic  solution  of  systems  of  equations. 

Ex.  1.    By  the  aid  of  loci  discuss  the  system 

ax  +  by  =  c,  (1) 

ax  +  b'y  =  c'.  (2) 


(a) 


Let  the  locus  of  (1)  be  the  straight  line  MN,  and  that  of  (2) 
the  line  liP.  Then  the  coor- 
dinates of  the  point  P,  which  is 
common  to  both  loci,  will  be 
the  solution  common  to  (1) 
and  (2),  or  the  solution  of  the 
system  (a).  By  measuring 
the  coordinates  OA  and  AP, 
the  numerical  solution  of  the 
system  could  be  obtained. 

This  example  illustrates 
graphically  the  theorem  in 
§207. 

The  loci  will  have  one,  and  only  one,  point  in  common, 
if  a/a'  ^  b/h>, 

i.e.,  if  (1)  and  (2)  are  independent  and  consistent  (§  207). 

The  loci  will  coincide  throvghotit  their  whole  extent, 
if  a/a'  ^  b/V  =  c/c', 

i.e.,  if  (1)  and  (2)  are  equivalent  (§§  207,  357). 

The  loci  will  be  parallel  and  have  no  point  in  common, 
if  a/a'  =  b/b'  and  a/a^  =^  c/c', 

i.e.,  if  (1)  and  (2)  are  inconsistent  (§§  207,  357). 


438 


ELEMENTS  OF  ALGEBBA 


Ex.  2.   By  the  aid  of  loci,  discuss  the  system 


for  different  values  of  c. 


x2  +  ?/2  =  25, 


(1) 

(2) 


(a) 


The  locus  of  (1)  is  the  circle  PP'BP"  ;  and,  if  c  =  1,  the  graph  of 
(2)  is  the  straight  line  MN;  hence  the  coordinates  of  the  two  points 
P  and  B  are  the  two  solutions  of  system  (a). 

By  measurement  we  find  the  two  solutions  to  be  3,  4  and  —  4,  —  3. 


As  c  increases,  the  locus  MN  moves  upward  parallel  to  itself,  and 
P  and  B  approach  P'. 

When  c  =  5\/2,  the  locus  of  (2)  is  the  tangent  N'3I',  and  the  two 
solutions  of  the  system  are  equal. 

Similarly,  when  c  =  —  5\/2,  the  locus  of  (2)  is  M"N". 

When  c  <  5\/2  and  >  -  5V2,  the  locus  of  (2)  lies  between  N'M' 
and  N"M",  and  the  two  solutions  of  the  system  are  real  and  unequal. 

When  c  >  5\/2  or  <  —  5V2,  the  locus  of  (2)  does  not  cut  the  circle, 
and  both  solutions  of  the  system  are  imaginary  or  complex. 


SOLUTION  OF  EQUATIONS  AND   SYSTEMS      489 


Ex.  3.    By  aid  of  loci  discuss  the  system 
x^+  y^  =  r^ 


(1) 


}(«) 


C2  +  ^2  _  ^2  (2) 

for  different  values  of  r. 

The  locus  of  (1)  is  the  ellipse  ABBS,  in  which  OA=S  and  0B=2. 


If  r  =  5/2,  the  locus  of  (2)  is  the  circle  PP'P"P"',  and  the  four 
solutions  of  the  system  are  the  coordinates  of  the  four  points  P,  P', 
P",  P"',  and  thus  are  real  and  unequal. 

If  r  =  3,  the  circle  will  be  tangent  to  the  ellipse  at  A  and  B ;  hence 
two  solutions  of  the  system  will  be  3,  0,  and  the  other  two  —  3,  0. 

If  r  =  2,  the  circle  will  be  tangent  to  the  ellipse  at  B  and  S. 

If  r  <  2  or  >  3,  the  two  loci  will  have  no  common  points,  and  all 
four  solutions  of  the  system  will  be  imaginary  or  complex. 

When  r  =  5/2,  by  clearing  (2)  of  fractions  and  then  subtracting  it 
from  (1)  we  obtain  5]/^  =  11,  the  locus  of  which  is  the  parallel  lines 
PP  and  P"'P'.  These  lines  cut  either  the  ellipse  or  the  circle  in  all 
the  points  which  are  common  to  these  curves,  and  only  in  these  points. 
This  illustrates  the  equivalency  of  system  (a)  to  the  system 


4x2  +  9y2  =  36, 
6y2 


=  36,  I         x'^-hy^  =  ^,] 
=  11.1    °^  5?/2=ll.  J 


440 


ELEMENTS  OF  ALGEBRA 


(a) 


Ex.  4.    By  aid  of  loci  discuss  the  system 

xy  =  12,  (1) 

y  =  mx  +   n,  (2)  , 

for  different  values  of  m  and  n. 

The  locus  of  (1)  is  the  curves  AB  and  02),  whose  infinite  branches 
approach  the  axes. 

When  n  =  0  and  m  =  3/4,  the  locus  of  (2)  is  the  line  PP',  and  the 
two  solutions  of  system  (a)  are  the  coordinates  of  the  points  P  and  P'. 

Let  m  =  0  ;  then  P  will  move  out  along  the  infinite  branch  PB,  and 
P'  along  the  infinite  branch  P'C  ;  that  is,  y  =  0  and  x  =  +  go  or  —  oo. 


y  lA 


Again,  when  n  =  0,  the  two  solutions  of  system  (a)  are 

2V:i/m,  2\/3hi,  and  -2\/^i/m,   -  2V8to. 
For  m  =  0,  the  solutions  in  (.3)  assume  the  forms 
a/0,  0  and  -  a/0,  0  ; 


(3) 


hence  equation  (1)  and  y  =  0  are  inconsistent,  and  system  (a)  is  then 
impossible.  This  agrees  with  the  figure,  for  the  locus  oiy  =  0  coincides 
with  X'OX,  and  does  not  intersect  the  locus  of  (1). 

When  m  is  negative,  the  solutions  in  (3)  become  imaginary.     This 
agrees  with  the  figure  ;  for  when  m  is  negative,  x  and  y  in  y  =  mx  are 


SOLUTION  OF  EQUATIONS  AND   SYSTEMS      441 

opposite  in  quality,  and  hence  the  locus  of  y  =  mx  will  lie  in  the 
second  and  fourth  quadrants,  and  will  not  cut  the  locus  of  (1). 

If  »i  =  0  and  n  ^  0,  the  locus  of  (2)  will  be  parallel  to  XOX',  and 
will  cut  the  locus  of  (1)  in  only  one  point ;  hence  system  (a)  will  be 
defective  in  one  solution. 

Exercise  149. 

1.  By  aid  of  loci  show  that  system  (a)  is  equivalent  to 
the  four  systems  in  (6). 

a^  -h  y-  =  25 
xy 


r  =  25 

x?/  =  12  J 


x  +  y  =  7]    x  +  y  =  7      )    x-^y  =  -7)     x-\-y  =  -7] 
a;  —  l  =  lJ    x  —  y  =  —  lj    x  —  y  =  \      i     x  —  y  =  —  l^ 

2.    By  aid  of  loci  show  that  the  following  six  systems  are 
equivalent : 


0^  +  2/2  =  25 

ar'=16 

0^  +  2/^  =  25 

7?-f  =  7    J 

2/2  =  9 

x'-f  =  7 

1                   ^-f=7 

a^  =  16 

a^  =  16 

1                  /  =  9           i 

/  =  9 

I 
1- 


GRAPHIC   SOLUTION  OF  EQUATIONS   IN   X. 


465.  A  variable  whose  value  depends  upon  one  or  more 
other  variables  is  called  a  dependent  variable,  or  a  function  of 
those  variables.  A  variable  which  does  not  depend  upon  any 
other  variable  for  its  value  is  called  an  independent  variable. 

E.g.,  x^,  2  a;"2  —  .3  X  +  7,  or  x"*  —  7  x^  +  9,  is  a  function  of  the  inde- 
pendent variable  x. 

Again,  y  in  each  of  the  equations  in  this  chapter  is  a  function  of  the 
independent  variable  x. 

The  symbol  f{x),  read  ^function  x,'  is  used  to  denote  any 
function  of  x. 


442  ELEMENTS   OF  ALGEBRA 

The  symbols  f(a),  f(2),  /(I)  represent  the  values  of  f(x) 
when  x  =  a,  2,  1,  respectively. 

E.g.,  iif(x)  =  ^3  +  X,  then 

/(a)  =a^^a,  /(2)  =  2^  +  2  =  10,  /(I)  =  2. 

Since /(a;)  denotes  any  function  of  x,  y  =f{x)  denotes  any 
equation  in  x  and  y,  when  the  equation  is  solved  for  y. 
Thus,  any  one  of  the  equations  in  the  first  ten  exsrcsples  in 
exercise  146  is  a  particular  case  of  y  =f(x). 

466.  A  continuous  variable  is  a  variable  which  in  passing 
from  one  value  to  another  passes  successively  through  all 
intermediate  values. 

A  function,  as /(a;),  is  said  to  be  continuous  between  x  =  a 
and  ic  =  6,  if  when  x  changes  continuously  from  a  to  h,  f(x) 
varies  continuously  from /(a)  to  f{b).  In  other  words, /(a:) 
is  continuous  between  x  =  a  and  x=b,  when  the  locus  of 
y  =f(x)  is  an  unbroken  curve  between  the  lines  x  =  a  and 
x  =  b. 

E.g.,  the  time  since  any  past  event  varies  continuously.  The  veloc- 
ity acquired  by  a  falling  body  and  the  distance  fallen  are  continuous 
functions  of  the  time  of  falling. 

In  each  of  the  four  examples  in  §  463,  y  is  a  continuous  function  of 
X  for  all  real  values  of  x. 

In  example  2  of  §  464,  y  in  equation  (1)  is  real  and  a  continuous 
function  of  x  between  x  =  —  6  and  x  =  +  5. 

The  examples  in  §  463  illustrate  the  fact  that 

Any  rational  integral  function  of  x  is  a  continuous  function. 

In  what  follows  we  shall  use  f(x)  to  denote  a  rational 
integral  function  of  x. 

467.  The  ordinates  of  the  points  in  the  locus  ot  y  =  a^ 
—  x  —  6  in  fig.  1,  of  §  463,  are  the  successive  values  of 
a^  —  x  —  6  corresponding  to  successive  values  of  x ;  hence, 
the  locus  of  y  =f{x)  is  often  caUed  the  graph  of  f{x). 


SOLUTION   OF  EQUATIONS  AND   SYSTEMS      443 

E.g.,  in  fig.  2,  while  x  increases  continuously  from  —3  to  zero, 
the  function  x^  —  x  —  6  decreases  continuously  from  +  6  though  zero 
to  —  6  ;  and  while  x  increases  from  zero  to  +4,  x^  —  x  —  Q  first  de- 
creases  from  —  6  and  then  increases  to  -f  6. 

Again,  in  fig.  3,  while  x  increases  continuously  from  —  2  to  —  0.8, 
the  function  x^  —  2x  increases  continuously  from  —  4  to  4- 1.1 ;  while 
X  increases  from  —0.8  to  +0.8,  x^  —  2x  decreases  from  +1.1  to 
—  1.1  ;  while  x  increases  from  +  0.8  to  +  2,  x'  —  2  x  increases  from 
-1.1  to  4. 

In  like  manner,  in  the  other  figures,  the  pupil  should  follow  the 
changes  in  f{x)  as  x  increases. 

468.  The  abscissas  of  the  points  in  which  the  graph  of 
f(x)  cuts  or  touches  the  axis  of  x  are  the  real  vahies  of  x 
for  which  f(x)  is  zero ;  that  is,  they  are  the  real  roots  of 
the  equation  f{x)  =  0. 

At  a  point  of  tangency  the  graph  is  properly  said  to  touch 
the  axis  of  x  in  two  coincident  j)oints. 

E.g.,  from  the  graph  in  fig.  2,  we  learn  that  one  root  of  the  equa- 
tion z^  —  x  —  Q  =  0  is  — 2  and  the  other  is  3. 

From  the  graph  in  fig.  3,  we  learn  that  the  three  roots  of  the  equa- 
tion ar^  —  2  X  =  0  are  —  ^^2,  0,  and  y/2. 

In  fig.  4,  the  graph  cuts  the  axis  of  x  at  (—  1,  0)  and  touches  it  at 
(2,  0)  ;  hence,  one  root  ofx''  —  3x2  +  4  =  0is  — 1  and  the  other  two 
roots  are  2  each. 

Hence,  to  find  the  real  roots  of  f(x)  =  0,  we  can  draw 
the  graph  of  f{x),  or  the  locus  of  y  =f(x),  and  measure  the 
abscissas  of  the  points  of  intersection  and  tangency  with 
the  ar-axis. 

Exercise  150. 

Construct  the  graph  of  f(x),  and  find  approximately  the 
real  roots  of  each  of  the  following  equations : 

1.  3^^x-2  =  0.  4.    x^-Sar-Ax-\-ll  =  0. 

2.  ar'-f  2a;-5  =  0.  5.    a^  -  4.0(^  -  6x  -  S  =  0. 

3.  .r^-3a;+-4  =  0.  6.    a;^- 4  a^-3a;+- 2  =  0. 


CHAPTER   XXXIV 
THEORY  OF  EQUATIONS 

489.   Horner's  method  of  synthetic  division. 

Let  it  be  required  to  divide 

Ax^  +  Bx^+  Cx-^  D  hj  X-  a. 

If  for  convenience  we  write  the  divisor  to  the  right  of  the  dividend 
and  the  quotient  below  it,  by  the  usual  method  we  have  : 

Ax^  +  Bx"^  +  Cx  -\-  D 

Ax^  —  Aax^ 


Ax'  +  (Aa  +  B)x 
+  (Aa^  +  Ba+  O) 


(Aa  +  B)x:^ 

(Aa  +  B)x^  -  (Aa^  +  Ba)x 

(Aa^  +  Ba  +  C)x 

{Acfi  +  Ba+  C)x  -  {Aa^  +  ga^  +  Ca) 

Aa^  +  Ba^  +  Ca  +  D 
In  the  shorter  or  synthetic  method,  we  write  only  the  coefficients  of 
the  dividend  and  place  a  at  their  right,  as  below  : 

A  B  G  D\a 

_  Aa  Aa^  +  Ba  Aa^  +  Bii^  +  Ca 

A  Aa-\-  B  Aa'^-h  Ba+C  Aa^  +  Ba"^  +  Ca -\-  D 

Multiplying  A  by  a,  writing  the  product  under  J?,  and  adding,  we 
obtain  Aa  +  B.  Multiplying  this  sum  by  a,  writing  the  product  under 
O,  and  adding,  we  obtain  Aa'^  +  Ba  +  C.  In  like  manner  the  last 
sum  is  obtained. 

Now  A  and  the  first  two  sums  are  respectively  the  coefficients  of 
x"^,  X,  and  x''  in  the  quotient  obtained  above  by  the  ordinary  method, 
and  the  last  sum  is  the  remainder. 

In  like  manner  any  rational  integral  function  of  x  can  be  divided 
by  X  —  a.  If  any  power  of  x  is  missing,  its  coefficient  is  zero,  and 
must  be  written  in  its  place  with  the  other  coefficients. 

Observe  that  the  shorter  or  synthetic  method  of  division 
includes  only  that  part  of  the  usual  method  given  above 
which  is  in  black-faced  type. 

444 


THEORY  OF  EQUATIONS  445 

Since  we  omit  the  sign  —  before  the  second  term  of  the 
divisor,  we  must  omit  also  that  sign  before  the  second  term 
of  each  product,  and  then  add  that  term  to  the  dividend,  as 
in  the  shorter  method  above. 

Here  the  remainder  Aa^  +  Bar  -{■  Ca -\-  D  is  the  value  of  the  divi- 
dend Ax^  +  Bx:^  +  Cx-t  D  for  x  =  A,  which  affords  a  second  proof 
of  §  131. 

Ex.  1.    Divide  2  x*  +  a;^  -  29  x2  -  9  x  +  180  by  x  -  4. 

"Write  the  coefiBcients  with  4  at  their  right  and  proceed  as  below  : 

2         +1         -29         -    9         +180|4 
4-8         +36         +28         +76 

2         +9        +7         +19         +256 

Hence  the  quotient  =  2  x**  +  9  x^  +  7  x  +  19, 
and  the  remainder,  or  /(4),=  256. 

Ex.  2.    Divide  2 x*  +  x^  -  29  x^  -  9x  +  180  by  x  +  5. 

2         +1         -29         -    9         + 180 1  -  5 
_  10         +45         -  80        +445 


2         -    9        +16         -89         +626 


Hence  the  quotient  =  2  x^  -  9  x^  +  10  x  -  89, 
and  the  remainder,  or  /(  -  5),  =  625. 

Ex.  3.   Divide  x^  +  21  x  +  342  by  x  +  6. 

1         +0         +21         +342 
_  0         +36         -  342 


1         _6         +57  0 

Hence  the  quotient  =  x^  —  6  x  +  57, 
and  the  remainder,  or  /(—  6),=  0. 

Hence  the  division  is  exact,  and  x  +  6  is  a  factor  of  /(x). 


Exercise  151. 
By  Horner's  method 

1.  Divide  x^ -2x^  -  4.X  +  8  by  x-3;  by  x  -  2. 

2.  Divide  2a;^  H-4a^-ar^- 16  a;  -  12  bya;-|-4;  by  x  +  3. 


446  ELEMENTS  OF  ALGEBRA 

3.  Divide  3  a;*  -  27  aj^  +  14  a?  +  120  by  x-%;  by  a;  +  5. 

4.  Find  the  value  of  2  a;*  —  3  a^^  +  3  a?  —  1  when  a;  =  4 ; 
when  a;  =  —  3 ;  when  a;  =  3  ;  when  x  =  5. 

5.  Show  that  one  factor  of  a;^  -f  8  x-^  +  20  a;  +  16  is  x  -{-2, 
and  from  the  quotient  find  the  others. 

6.  Show  that  two  factors  of  a^*  +  a^  -  29  a;^  -  9  a;  +  180 
are  x  —  3  and  a;  -f  3,  and  find  the  others. 

7.  Show  that  two  factors  of  a?^—  4  ar^—  8  a;  +  32  are  x  —  2 
and  a;  —  4,  and  find  the  others. 

INTEGRAL   RATIONAL   EQUATIONS   IN   ONE   UNKNOWN. 

470.  If  all  the  terms  of  an  integral  rational  equation  in  x 
are  transposed  to  the  first  member  and  arranged  in  descend- 
ing powers  of  x,  we  shall  obtain  an  equivalent  equation  of 
the  form 

Aa^"  +  ^1^"-'  +  ^l2^"-'  +   •  •  •  +  ^n-l^  +  -4n  =  0,  (B) 

where  Aq,  Ai,  A2,  •••,  A^-i,  A^  denote  any  known  numbers, 
real,  imaginary,  or  complex,  and  n  denotes  the  degree  of  the 
equation. 

Denoting  the  first  member  of  (B)  by  f(x),  (B)  can  be 
written 

471.  To  solve  equation  (B),  or  f(x)  =  0,  by  §  149  we 
need  to  factor  its  first  member,  equate  each  factor  to  zero, 
and  solve  the  resulting  equations.  But  when  (JB)  is  above 
the  second  degree  in  x,  the  first  member  cannot  be  factored 
by  inspection  except  in  certain  special  cases. 

The  methods  which  follow  should  be  used  when,  and  only 
when,  f(x)  cannot  be  factored  by  inspection. 

472.  If  a  is  a  root  of  the  equation  f(x)  =  0,  that  is,  if 
f{a)  =  0,  theii  f{x)  is  divisible  by  x—a  (§  131). 


THEORY  OF  EQUATIONS  447 

Conversely,  if  f(x)  is  divisible  by  x  —  a,  then  f(a)  =  0 ;  tJiat 
is,  a  is  a  root  of  the  equation  f{x)  —  0. 

E.g.^  if  2  is  a  root  of  the  equation 

a;3_2x-2-4a;  +  8  =  0,  (1) 

then  its  first  member  is  divisible  by  x  —  2  (§  132). 

Conversely,  if  the  first  member  of  (1)  is  divisible  by  a;  —  2,  then  2  is 
a  root  of  this  equation. 

473.  It  was  proved  in  §  148  that  n  linear  equations  in  x 
are  jointly  equivalent  to  an  equation  of  the  ?ith  degree  in  x. 

In  proving  the  converse  of  this  theorem  in  the  next  article 
we  assume  the  following  theorem  : 

Any  integral  rational  equation  in  one  unknown  has  at  least 
one  root,  real,  imaginary,  or  complex. 

Note.     The  proof  of  this  theorem  is  too  long  and  difficult  to  be 
given  here. 

474.  Any  equation  of  the  nth  degree  in  one  unknown  has  n, 
and  only  n,  roots. 

Pi'oof     By  §  473,  the  equation  f{x)  —  0  has  a  root. 
Let  rtj  denote  this  root ;  then,  by  §  472,  f{x)  is  divisible 
by  a;  —  Oj,  so  that 

f{x)  =  {:x-a,)f{x),  (1) 

in  which,  by  the  laws  of  division,  f  (x)  has  the  form  of  f{x), 
and  is  of  the  {n  —  l)th  degree. 

Now  the  equation  f  (x)  =  0  has  a  root. 

Denote  this  root  by  a^ ;  then 

f{x)  =  (x-a.;)f(x),  (2) 

in  which  f2(x)  is  of  the  (?i  —  2)th  degree. 

Repeating  this  process  n  —  1  times,  we  finally  obtain 

fn-l(x)  =  (^-Ctn)A^,  (n) 

where  A^  is  the  coefficient  of  x''  in  f(x). 


448  ELEMENTS   OF  ALGEBBA 

From  (1),  (2),  •••,  {n),  we  obtain 

f(x)  =  {x-a,)f^{x) 

=  (x-a,)(x-ao)f2(x) 

=  (x~  ai)  (x  —  as)  (^*  —  ttg)  •  •  •  (a?  —  a„)  A-  (3) 

Hence  the  equation  f(x)  =  0  is  equivalent  to  the  n  linear 
equations 

X  —  ai  =  Oy   a;  —  0-2  =  0,  •  •  •,  x  —  «„  =  0, 

and  therefore  has  n  and  only  n  roots. 

From  (3),  it  follows  that  any  expression  of  the  71th.  degree 
in  X  can  be  resolved  into  n  linear  factors  in  x. 

475.  Equal  roots.  If  two  or  more  of  the  factors  x  —  a^, 
X  —  a2,  '•',  X  —  a^  are  equal,  the  equation  /(x)  =  0  has  two 
or  more  equal  roots. 

E.  g. ,  of  the  equation 

(x-4)3(a;  +  5)2(a;-7)  =  0, 
three  roots  are  4  each,  and  two  are  —  5  each. 

Ex.    One  root  of  2  ic^  -  5  x^  -  37  x  +  60  =  0  is  5.     Find  the  others. 
One  root  being  5,  one  factor  of /(oj)  is  x  —  5  (§  472). 
By  division  the  other  factor  is  found  to  be  2  x^  +  5  x  —  12. 
Hence  the  two  roots  required  are  those  of  the  equation 

2x2  +  5x-12^0.  (1) 

The  roots  of  (1)  are  evidently  —  4  and  3/2. 

Exercise  152. 
Solve  each  of  the  following  equations : 

1 .  cc^  —  6  £c^  +  10  a;  —  8  =  0,  one  root  being  4. 

2.  3  a^  —  25  .^•2  +  42  ic  4-  40  =  0,  one  root  being  5. 

3.  2  ic^  4-  a;^  —  15  a;  —  18  =  0,  one  root  being  —  2. 

4.  3  a^  -  8  a;2  -  31  a;  +  60  =  0,  one  root  being  -  3. 


THEORY  OF  EQUATIONS  449 

5.  4  a^  -  9  x^  —  3  a;  -h  10  =  0,  one  root  being  —  1. 

6.  a;^  +  a^  -  29  x2  -  9  X  4- 180  =  0,  two  roots  being  3  and 
-3. 

7.  a;''  —  4a;^  —  8x4-32  =  0,  two  roots  being  2  and  4. 

8.  2  x^  —  15  x^  4-  35  x^  —  30  a;  +  8  =  0,  two  roots  being  1 
and  2. 

9.  3  a;^  —  5  a;''  -  17  ^2  4-  13  X  4-  6  =  0,  two  roots  being  —2 
and  3. 

By  §  148,  form  the  equation  whose  roots  are : 

10.  The  two  numbers,  ±  V— 2. 

11.  The  four  numbers,   ±V— 3,   ±  V— 5. 

12.  The  four  numbers,  3  ±  V^^,  5,  —  2/3. 

13.  3/4,  liV^Ts,  l-t-V^=^. 

14.  2,   ±V^^,  3±V^^. 

15.  3,  -4,  V^^. 

In  each  of  the  last  six  examples,  observe  that  the  coefficients  of  the 
equation  obtained  are  all  real  when,  and  only  when,  the  imaginary  or 
complex  roots  occur  in  conjugate  pairs.  This  illustrates  the  converse 
of  the  next  article. 

476.  In  any  integral  rational  equation  having  only  real 
coefficients^  imaginary  or  complex  roots  occur  in  conjugate 
pairs;  that  is,    if  a  4-  bi  is  a  root,  then  a  —  bi  is  also  a  root. 

Proof.  If  the  coefficients  in  f(x)  are  all  real,  then  all  the 
terms  of  the  expression  obtained  by  substituting  a  4-  hi  for 
X  in  fix)  will  be  real  except  those  containing  odd  powers  of 
hi,  which  will  be  imaginary. 

Representing  the  sum  of  all  the  real  terms  by  A,  and  the 
sum  of  all  the  imaginary  terms  by  Bi,  we  have 

f{a  +  hi)  =  AJrBi.  (1) 


450  ELEMENTS   OF  ALGEBRA 

Now  f(a  —  bi)  will  evidently  differ  from  f(a  +  bi)  only  in 
the  signs  before  the  terms  containing  the  odd  powers  of  bi ; 
that  is,  in  the  sign  before  Bi ;  hence 

f(a  -  bi)  =  A-  Bi.  (2) 

Since  a  +  bi  is  a  root  of  f(x)  —  0,  from  (1)  we  have 

A  +  Bi  =  0. 

Therefore  ^  =  0  and  ^  =  0.  §  279 

Hence  by  (2),  f(a  -  bi)  =  0. 

That  is,  when  a  -\-  bi  is  a  root  of  f(x)  =  0,  a  —  bi  is  also  a 
root. 

Ex.     One  root  oix^  -  4x^  +  4x  -  S  =  0  (1) 

is  (1  +  V^3)/2  ;  find  the  others. 

Since  1/2  +  aA^/2  is  a  root,  1/2  -  V^^/2  is  also  a  root  (§  476). 
Hence  two  factors  of  the  first  member  of  (1)  are 

X  -  1/2  -  V^/2  and  X-1/2+  V^^/2, 

whose  product  is  (x  —  1/2)^  +  3/4,  or  x^  —  x+1. 

But  cc3  -  4ic-2  +  4rB  -  3  =  (x2  -  ic  +  1)  (a;  -  3);  (2) 

hence  the  third  root  of  (1)  is  3. 

Identity  (2)  illustrates  the  following  principle : 

477.  Any  rational  integral  function  of  x  wliose  coefficients 
are  real  can  be  resolved  into  real  factors,  linear  or  quadratic 
in  X. 

Proof.  If  the  coefficients  of  f{x)  are  real,  the  imaginary 
or  complex  roots  of  f{x)  =  0  occur  in  conjugate  pairs,  as 
a  4-  bi  and  a  —  bi-^  hence  the  complex  factors  of  f{x)  occur 
in  conjugate  pairs,  as  x  —  a  —  bi  and  x  —  a-{-  bi,  whose 
product  is  a  real  quadratic  expression  in  x-,   that  is 

(x  —  a—bi)  (x  -  a  H-  bi)  =  {x  -  ay  -|-  b^. 


THEORY  OF  EQUATIONS  451 

Exercise  153. 

Solve  each  of  the  following  equations,  and  find  the  real 
factors  of  the  first  member : 

1.  a^  -  6  ar'  +  57  a;  - 196  =  0,  one  root  being  1  -  4 V^3. 

2.  a:^  —  6  a;  +  9  =  0,  one  root  being  (3  +  V^^)/2. 

3.  ar'  -  2  a^  +  2  a;  -  1  =  0,  one  root  being  (1  +  V^r3)/2. 

4.  a;*-f4aj^  +  5ar^-|-2a;  —  2  =  0,  one  root  being  —  1  +  ^. 

5.  a;*4-4a^4-6a^  +  4a;  +  5  =  0,  one  root  being  i. 

6.  a^  —  ar*  4-  a;''  —  a.*^  +  a;  —  1  =  0,  two  roots  being  —  i  and 
(l  +  V^/2. 

7.  Show  that  in  an  equation  with  commensurable  real 
coefficients,  surd  roots  occur  in  conjugate  pairs;  that  is,  if 
a  +  -y/b  is  a  root  of  f(x)  =  0,  a  —  ->/6  is  a  root  also,  -y/b  being 
a  surd  number. 

All  the  terms  in /(a  +  ^/b)  will  be  rational  except  those  containing 
odd  powers  of  -^6,  whiph  are  surd.  Denote  the  sum  of  all  the  rational 
terms  by  A  and  the  sum  of  all  the  surd  terms  hy  B^b  ;  then 

/(a+  y/b)  =  A  +  By/b. 

Hence  /(a  —  y/b)  =  A  —  By/b;  and  so  on  as  in  §  476. 

8.  Solve  6  a^  — 13a;^  — 35  ar^  —  a;  + 3  =  0,  one  root  being 
2-V3. 

9.  Solve  X*-  36  a^+  72  a;  -  36  =  0,  one  root  being  3  -  V3. 

478.  The  graph  of /(»)  illustrates  the  fact  that  equal  real 
roots  form  the  connecting  link  between  unequal  real  roots 
and  imaginary  or  complex  roots,  and  that  imaginary  or  com- 
plex roots  occur  in  pairs. 

E.g.,  by  slightly  diminishing  the  term  4  of  the  function  x^—Sx^-\-i, 
its  graph  in  fig.  4  of  §  46.3  would  be  moved  downward,  and  would 
then  cut  the  axis  of  x  in  three  points  ;  by  slightly  increasing  the  term 


452  ELEMENTS   OF  ALGEBRA 

4,  the  graph  would  be  moved  upward,  and  would  then  cut  the  axis  of  x 
in  hut  one  point. 

That  is,  the  two  equal  real  roots  of  the  equation 

x3  _  3  x2  +  4  =  0 

would  become  unequal  real  roots  or  complex  roots  according  as  the 
known  term  4  were  diminished  or  increased. 

From  fig.  5  in  §  463  the  pupil  should  follow  the  changes  in  the 
roots  of  the  equation 

ic4  +  x3  -  3  x2  -  a;  +  2  =  0, 

(i)  when  the  term  2  is  decreased  continuously  to  -  1 ; 
(ii)  when  the  term  2  is  increased  continuously  to  4. 

479.  An  equation  of  the  form  (B)  in  §  470,  is  said  to  be 
in  the  type-form  when  the  coefficient  of  ic"  is  1. 

E.g.,  ic4  _  I  a;3  +  3  ^.2  _^  4  _  0  is  in  the  type-form. 

480.  If  an  equation  of  the  nth  degree  is  in  the  type  form,  then 

—  the  coefficient  of  x""'^  =  the  sum  of  the  roots; 

the  coefficient  of  x""'^  =  the  sum  of  the  products  of  the 
roots  taken  two  at  a  time; 

—  the  coefficient  of  jr"~^  =  the  sum  of  the  products  of  the 

roots  taken  three  at  a  time. 

(—  1)**  (the  coefficient  of  x^)  =  the  product  of  the  n  roots. 

Proof.     Let  a^,  a2,  a^,  •••  a„  denote  the  n  roots;  then,  by 
§  148,  the  equation  can  be  written  in  the  form 

(x  —  tti)  (x  —  ag)  (x  —  as)"'{x—  a„)  =  0.  (1) 

When  n  =  2,  by  multiplication  (1)  becomes 
a^  —  (ai  +  ttg)  ^  +  cti<^2  =  0, 
which  proves  the  theorem  when  n  =  2. 

When  71  =  3,  by  multiplication  (1)  becomes 

^  —  («i  +  «2  4-  cis)^  +  (<^i«2  +  «i0t3  +  a2a3)x  —  aia^^  =  0,    (2) 
which  proves  the  theorem  when  ?i  =  3. 


THEORY  OF  EQUATIONS  45B 

From  the  laws  of  multiplication  it  is  evident  that  the 
same  relation  holds  when  w  =  4,  5,  6,  •••. 

Observe  that,  if  the  term  in  a;""^  is  wanting,  the  sum  of 
the  roots  is  0,  and  ?.f  the  kno^n  term  is  wanting,  at  least 
one  root  is  0. 

E.g.,  in  the  equation 

x4  +  6«2_iix-6  =  0, 

the  sum  of  the  roots  is  0  ;  the  sum  of  their  products  taken  two  at  a 
time  is  6  ;  the  sum  of  their  products  taken  three  at  a  time  is  11 ;  and 
their  product  is  —  6. 

Note.  The  coeflBcients  in  any  equation  are  functions  of  the  roots  ; 
and  conversely,  the  roots  are  functions  of  the  coefficients.  The  roots 
of  a  literal  quadratic  equation  have  been  expressed  in  terms  of  the 
coefficients  (§  291).  The  roots  of  a  literal  cubic  or  biquadratic  equa- 
tion can  also  be  expressed  in  terms  of  the  coefficients,  as  is  shown  in 
college  algebra.  But  the  roots  of  a  literal  equation  of  the  fifth  or 
higher  degree  cannot  be  so  expressed,  as  was  proved  by  Abel  in  1825. 

Ex.   Its  roots  being  in  arithmetic  progression,  solve 

4  x3  -  24  x=«  +  23  X  +  18  =  0.  (1) 

Let  a  denote  the  second  term  in  the  A.  P.  and  b  the  difference  ; 
then  the  three  roots  are  a  —  b,  a,  a  +  b.  Hence  their  sum  is  3  a  ; 
the  sum  of  their  products  taken  two  at  a  time  is  3  a^  —  b'^  ;  and  their 
product  is  a{a^  —  b^). 

Divide  (1)  by  4  to  reduce  it  to  the  type-form;  then,  by  §480, 
we  have 

3  a  =  6,  3  a2  -  62  =  23/4,  a(a^  -b^)  =  -  9/2.  (2) 

Solving  the  first  two  equations  in  (2),  we  obtain  a  =  2,  b  =  ±  5/2  j 
and  these  values  are  found  to  satisfy  the  third  equation  in  (2). 
Hence  the  roots  are  —  1/2,  2,  and  9/2. 

Exercise  154. 

1.  The  sum  of  two  of  its  roots  being  zero,  solve 

The  sum  of  the  three  roots  is  —  4  ;  hence  the  third  root  is  —  4. 

2.  Its  roots  being  in  arithmetic  progression,  solve 


454  ELEMENTS  OF  ALGEBRA 

3.  Its  roots  being  in  geometric  progression,  solve 

3  0^3  _  26  a^  +  52  a:  -  24  =  0. 

4.  One  root  being  1  —V—  3,  solve 

aj3-4«2^g^_3^Q^ 

One  root  being  1  —  V— 3,  a  second  root  is  1  +  V—  3. 
The  sum  of  these  two  roots  is  2,  and  the  sum  of  the  three  roots 
is  4  ;  hence  the  third  root  is  2. 

5.  By  §  480,  solve  each  of  the  first  five  examples  in 
exercise  153. 

481.  If  the  coefficients  of  /(a?)  are  all  +,  /(a;)>  0  when 
a;>0;  hence,  if  the  coefficients  off{x)  are  all  positive,  f(x)  =  0 
has  no  positive  real  root. 

If  the  coefficients  of  f(x)  are  alternately  +  and  —  ;  then, 
when  a;<0,  /(aj)>0  or  <0  according  as  n  is  even  or  odd; 
hence,  if  the  coefficients  of  f{x)  are  alternately  4-  a7id  — , 
f  (jr)  =  0  has  no  negative  real  root. 

If  the  sum  of  the  coefficients  of  f(x)  is  zero,  /(I)  =  0 ; 
hence,  when  the  sum  of  the  coefficients  of  f(x)  is  zero,  one  root 
of  f(x)  =  Ois  -\- 1. 

E.g.,  x^  +  6  x^  -\-  llx  -\-  6  =  0  has  no  positive  root,  since  /(x)>  0 
when  OS  >  0. 

o;^  —  6  a;2  _|-  10  X  —  8  =  0  has  no  negative  root ;  since  /(x)<  0  when 
aj<0. 

/k4  +  2  ic3  -  13  ic2  -  14  X  +  24  =  0  has  +  1  as  a  root ;  since  /(I)  =  0. 

482.  If  all  the  coefficients  of  an  equation  in  the  typeform 
are  whole  numbers,  any  commensurable  real  root  of  the 
equation  is  an  integral  factor  of  its  known  term. 

E.g.,  any  commensurable  real  root  of  the  equation 

a;3  _6a;2  +  i0a;-8  =  0 

is  an  integral  factor  of  its  known  term  —  8  ;  that  is,  any  such  root  is 

±1,  ±2,  ±4,  or  i8. 


THEORY  OF  EQUATIONS  455 

Proof.    Let  all  the  coefficients  of  the  equation 

a-"  +  Aix""-^  +  A^x""-^  -i \-A^  =  0  (1) 

be  whole  numbers,  and  suppose  that  s/t,  a  fractional  num- 
ber in  its  lowest  terms,  is  one  of  its  roots. 
Substituting  s/t  for  x,  we  obtain 

oft  ©n-l  ©n— 2 

Multiplying  by  r~^,  and  transposing,  we  obtain 

s^^/t  =  -  (Ais^-^  +  A^s""-^  +  •  •  •  4-  A„r-^)  (2) 

Now  (2)  is  impossible,  for  its  first  member  is  a  fractional 
number  in  its  lowest  terms,  and  its  second  member  is  a 
whole  number. 

Hence  a  fractional  number  cannot  be  a  root,  and  there 
fore  any  commensurable  root  must  be  a  whole  number. 

Next,  let  a  be  an  integral  root  of  (1). 

Substituting  a  for  ic,  transposing  A^,  and  dividing  by  a, 
we  have 

a"-i  +A,a--''  +  ^,a-«  +  •  •  •  +  A,,_,  =  -  AJa.         (3) 

The  first  member  of  (3)  is  a  whole  number;  hence  the 
quotient  AJa  is  a  whole  number,  ^.e.,  a  is  an  integral 
factor  of  A^. 

Ex.1.     Solve  a;8  -  6x2 +  10x- 8  =  0.  (1) 

By  §  481,  (1)  has  no  negative  root ;  hence,  by  §  482,  any  commen- 
surable real  root  of  (1)  is  +  1,  +  2,  +  4,  or  +  8,  i.e.  it  is  one  of  the 
positive  integral  factors  of  8. 

The  work  of  determining  whether  -)-  4  is  a  root  can  be  arranged 

as  below : 

1         _6         +10  -S[± 

+4         -    8  +8 

1-2+2  0 

The  division  is  exact,  and  the  quotient  is  x^  —  2  a;  +  2. 
Hence  the  roots  of  (1)  are  4  and  the  roots  of 

x2-2x  +  2  =  0.  (2) 


456  ELEMENTS  OF  ALGEBRA 

Solving  (2),  x  =  l  ±  V^. 

Hence  the  roots  of  (1)  are  4  and  1  ±  V—  1. 

Ex.2.     Solve  a;*  +  2  ic3  -  13x2- 14^  +  24  =  0.  (1) 

By  §  481,  one  root  of  (1)  is  +  1,  and  by  §  482  any  other  commen- 
surable real  root  is 

±1,   ±2,   ±3,   ±4,   ±6,   ±8,   ±  12,  or  ±  24, 

i.e.  it  is  one  of  the  integral  factors  of  24. 


1 

+  2 
+  1 

-13 

+    3 

-14 
-10 

+  24|J_ 
-24 

1 

+  3 
-2 

-10 

-    2 

-24L 
+  24 

-2 

1         +1         -  12 

Hence  the  roots  of  (1)  are  1,  —  2,  and  the  roots  of 

a;2  +  a;  -  12  =  0. 

Hence  the  roots  of  (1)  are  1,  —  2,  3,  and  —  4. 
Usually  it  is  better  to  try  the  smaller  factors  of  An  first. 

Exercise  155. 
Solve  each  of  the  following  equations : 

1.  i^  +  2x'  +  9x-{-lS  =  0. 

2.  a^-6a^-{-llx-6  =  0. 

3.  a^-4.x^-6x-{-9  =  0. 

4.  x^-3a^-{-x'-^2x  =  0. 

5.  x^-Sa^-\-13x-6  =  0. 

6.  aj3  +  6a;2-f-9aj  +  2  =  0. 

7.  a^  +  5a^-9a;-45  =  0. 

8.  x^-Aa^-Sx-^32  =  0. 

9.  x*-6x^  +  2Ax-16  =  0. 

10.  x^-SiK^-Ux'-^4Sx-32  =  0. 

11.  a^-3x*-9a^-\-21x'-10x-\-24.  =  0. 

12.  a^-\-2x^-23x-60  =  0. 


THEOBY  OF  EQUATIONS  467 

483.  Limits  of  real  roots.  Superior  limit.  In  evaluating 
/(4)  in  example  1  of  §  469,  the  sums  are  all  positive,  and 
they  evidently  would  all  be  greater  for  x  >  4. 

Hence  f{x)  can  vanish  only  for  x  <  4 ;  and  therefore  all 
the  roots  of  f(x)  =  0  are  less  tlian  4. 

Hence,  if  in  computing  the  value  of  f('^c)  all  the  sums  are 
positive,  the  real  roots  off(x)  =  0  are  all  less  than  +(?. 

The  least  integral  value  of  +c  which  fulfils  this  condition 
is  called  the  superior  limit  of  the  real  roots  of  f(x)  =  0. 

Inferior  limit.  In  evaluating  /(— 5)  in  example  2  of 
§  469,  the  sums  are  ^ternately  —  and  -f,  and  they  evidently 
would  all  be  greater  arithmetically  for  x<  —  5.  Therefore 
all  the  real  roots  of  f(x)  =  0  are  greater  than  —  5. 

Hence,  if  in  computing  the  value  of  f{~b)  the  sums  are  alter- 
nately —  and  -}-,  all  the  real  roots  of  f{x)  =  0  are  greater 
than  ~b. 

The  greatest  integral  value  of  ~h  which  fulfils  this  condi- 
tion is  called  the  inferior  limit  of  the  real  roots  of  f(x)  —  0. 

Observe  that  the  above  reasoning  holds  when  we  regard  a 
zero  sum  as  either  positive  or  negative,  and  that  when  the 
last  sum  is  zero,  the  limit  obtained  is  itself  a  root. 

E.g.,  if       /(x)  =  x*  +  2x8- 13a:2-14a;  +  24  =  0;  (1) 

then  in  evahiating/(4),  the  sums  are  all  + ;  and  in  evaluating/(—  5), 
the  sums  are  alternately  —  and  + ;  hence  the  real  roots  of  f(x)  =  0  lie 
between  —  5  and  4. 

Hence,  by  §  482,  any  commensurable  roots  of  (1)  must  be 

±1,   ±2,   ±3,  or  -4. 

Compare  this  result  with  example  2  in  §  482. 

Exercise  156. 
1.    Show  that  any  commensurable  real  root  of 
a:3_2a:_50  =  0 
lies  between  —  2  and  4 ;  and  hence  is  ±  1  or  2. 


458  ELEMENTS  OF  ALGEBRA 

2.  Show  that  any  commensurable  real  root  of 

is   ±  1,   ±  2,   ±  4,   ±  5,   ±  8,  or  10. 

3.  Show  that  any  commensurable  real  root  of 

a;4_i5a^-|.10a;  +  24  =  0  (1) 

is   ±1,   ±2,   ±3,  or  -4. 

4.  Eind  the  roots  of  equation  (1)  in  example  3. 

Solve  each  of  the  following  equations: 

5.  x*-9ic^  +  lTa;'^+27a;-60  =  0. 

6.  «^-45a^-40x  +  84  =  0. 

7.  a^_4a;*-16i»«  +  112a;2-208a;  +  128  =  0. 

8.  ic4-ar^~39a;2  +  24ic  +  180  =  0. 

9.  a^  +  5ar^  -  81  a;^  -  85 ar^  +  964a^  -f  780aj  - 1584  =  0. 

10 .  ic^  +  aj«  -  14  a^  -  14  a;^  4-  49  aj^  +  49  a^  -  36  a;  =  36. 

11.  aj6-10x4-3a.-2  +  108  =  0. 

12.  a;«-2ar^-7a;*  +  20aj3-21a^-18a;-f-27  =  0. 

484.  To  transform  an  equation  into  another  whose  roots 
shall  be  some  multiple  of  those  of  the  given  one. 

Proof.     If  in  the  equation 

of"  +  ^ix"-i  +  A^x^-^  +  ^3X"-3  H h  ^„  =  0,        (B) 

we  put  X  =  Xi/a,  and  multiply  by  a",  we  obtain 

a;i«  +  ^laa^i"-^  +  ^gOt V~'  +  ^sO^V"^  +  •  •  •  +  -4^  =  0.  (2) 

Since  Xi  =  aa;,  the  roots  of  (2)  are  a  times  those  of  (1). 

Hence,  to  effect  the  required  transformation,  multiply  the 
second  term  of  (B)  by  the  given  factor,  the  third  term  by  its 
square,  and  so  on. 

Observe  that  before  the  rule  is  applied  the  equation  must 


THEORY  OF  EQUATIONS  459 

be  put  in  the  type-form,  and  any  missing  power  of  x  must 
be  written  with  zero  as  its  coefficient. 

This  theorem  becomes  evident  also  when  we  observe  that 
if  in  equation  (2)  in  §  480  each  root  is  multiplied  by  a,  the 
second  term  will  be  multiplied  by  a,  the  third  term  by  a^, 
and  the  fourth  term  by  a\ 

The  chief  use  of  this  transformation  is  to  clear  an  equa- 
tion of  fractional  coefficients. 

Ex.   Solve  the  equation 

x3-.i^x2  +  fx-x^^  =  0,  (1) 

first  transforming  it  into  another  with  integral  coefficients. 

Multiplying  the  second  term  by  a,  the  third  by  a-,  the  fourth  by  a^, 

we  obtain 

x3  _  J^  ax2  -f  f  a^x  -  i^«  a'  =  0.  (2) 

By  inspection  we  discover  that  4  is  the  least  value  of  a  which  will 
render  the  coefficients  of  (2)  integral.    Putting  a  =  4,  we  obtain 

a;8  _  11  x2  +  36  X  -  36  =  0.  (3) 

The  roots  of  (3)  are  found  to  be  2,  3,  and  6. 

But  the  roots  of  (3)  are  four  times  the  roots  of  (1);  hence  the  roots 
of  (1)  are  1/2,  3/4,  and  3/2. 

Exercise  157. 

Solve  the  following  equations  by  transforming  them  into 
others  whose  commensurable  real  roots  are  whole  numbers : 

1.  a^-^x^-^^ix  +  ^\  =  (). 

2.  a^-x'/4:-x/2-\-l/S  =  0. 

3.  8a^-26a;2-hllx-hl0  =  0. 

4.  a^-  x^/3  -  x/m  +  1/108  =  0. 

5.  24:3^ -520^ +  26x-S=-0. 

6.  9x*-9a^  +  5x'-3x-^2/S  =  0. 

7.  x'  -  ^76  -  ar^/12  -  13  a^/24  +  1/4  =  0. 

8.  2a;^-12ar^  +  19a:'-6a;4-9  =  0. 


460  ELEMENTS   OF  ALGEBRA 

485.  If  f(a)  and  f(b)  are  opposite  in  quality,  an  odd  number 
of  real  roots  off(x)  =  0  lies  between  a  and  b. 

Iffip)  and  fib)  are  like  in  quality,  no  real  root,  or  an  even 
number  of  real  roots  of  f(x)  =  0  lies  between  a  and  b. 

Proof  If  the  ordinates  of  two  points  in  the  graph  of  fix) 
are  opposite  in  quality,  the  points  are  on  opposite  sides  of 
the  a^axis,  and  the  part  of  the  graph  between  these  points 
must  cross  that  axis  an  odd  number  of  times  (§  466) ;  that 
is,  f{x)  is  zero  for  an  odd  number  of  values  of  x  between  a 
and  6. 

If  the  ordinates  of  two  points  are  like  in  quality,  the 
points  are  on  the  same  side  of  the  avaxis,  and  the  part  of 
the  graph  between  these  points  either  does  not  cross  that 
axis  or  crosses  it  an  even  number  of  times,  touching  it  being 
regarded  as  crossing  it  twice. 

E.g.,  in  fig.  3  of  §  463,  the  graph  cuts  XJT  an  odd  number  of 
times  between  A  and  B  or  A  and  Z>,  and  an  even  number  of  times 
between  A  and  C  or  B  and  D. 

In  fig.  5  of  §  463,  the  graph  cuts  XXf  an  odd  number  of  times 
between  A  and  B  or  B  and  E,  and  an  even  number  of  times  between 
A  and  (7,  C  and  E,  or  A  and  E. 

Ex.    Find  the  first  figure  of  eacli  real  root  of  the  equation 

ic3  _  4  a;2  _  6  ic  +  8  =  0.  (1) 

By  §§  474  and  476,  (1)  has  either  three  or  only  one  real  root. 
By  Horner's  method  we  find  that :  - 
when  x  =  -2,   -1,       0,        1,         2,         3,         4,        5, 

f(x)=-4,   +9,   +8,   -1,   -12,   -19,   -16,  +3. 

Since  /(—  2)  and  /(—  1)  are  opposite  in  quality,  at  least  one  root 
of  (1)  lies  between  —  2  and  —  1.  For  like  reason  a  second  root  lies 
between  0  and  1,  and  a  third  between  4  and  6. 

Hence  two  roots  are  -(!•  +  )  and  4-  +  ,  and,  since  /(0.9)  is  + 
and  /(I)  is  -,  the  third  root  is  0.9  +. 


THEORY  OF  EQUATIONS  461 

486.  Any  equation  of  an  odd  degree  in  which  Aq  is  positive 
has  at  least  one  real  root  whose  quality  is  opposite  to  that  of 
its  known  term  A^. 

Proof   If  J-o  >  0  and  f{x)  is  of  an  odd  degree,  then 

/(-^)  is  -,  /(O)  =  A,  /(+^)  is  +. 

Hence  if  A^  is  positive,  one  root  of  f(x)  =  0  lies  between 
—00  and  0  (§  485) ;  and  if  A^  is  negative,  one  root  lies 
between  0  and  +qo. 

487.  Any  equation  of  an  even  degree  in  which  A^  is  positive 
and  the  known  term  A^  is  negative  has  at  least  one  positive 
and  one  negative  real  root. 

Proof     If  A  >  0  and  f{x)  is  of  an  even  degree,  then 

/(-oo)  is  +,  /(O)  is  -,  X+^)  is  +. 
Hence  one  root  of  f{x)  =  0  lies  between  —  oo  and  0,  and 
another  between  0  and  +qo. 

Exercise  158. 
Find  the  first  figure  of  each  real  root  of  the  equations : 

1.  a^-3x2-4a:4-ll  =  0.     5.    a^-2aj-5  =  0. 

2.  a^  +  a^-2x-l  =  0.  6.    a:^  ^_  a;  _  50O  =  0. 

3.  x*-4:2i^-3x-\-2S  =  0.      7.    x^ -\- 10  x"  +  5  x  =  260. 

4.  x^-4:X--6x  =  -S.  8.    a;*-12a.-2+12a:-3  =  a 


48  FRENCH. 


The  Qa  Ira  Series  of  French  Plays. 

Edited  by  Professor  B.  W.  Wells,  of  the  University  of  the  South. 
6  volumes,  i6mo,  cloth.    Each,  36  cents. 

THE  plays  selected  have  not  heretofore  been  edited  for  use  in 
American  schools.  The  series  contains  works  acfapted  to 
the  most  varied  needs,  but  these  works  are  so  treated  as  to  ex- 
clude all  expressions  or  suggestions  which  could  stand  in  the  way 
of  their  use  in  mixed  classes.  The  introductions  give  brief 
biographies  of  the  authors  and  such  comment  on  their  work  as 
may  seem  helpful.  The  Notes  explain  peculiarities  of  idiom  that 
would  not  naturally  be  sought  in  a  dictionary.  Allusions  to 
social  and  political  customs,  as  well  as  to  literature  and  history, 
receive  such  comment  as  will  aid  the  pupil  to  put  himself  in  the 
place  of  the  original  audience.  In  this  way  it  is  hoped  that  the 
reading  of  these  plays  will  help  the  student  not  only  in  the  study 
of  French,  but  also  in  the  development  of  a  literary  taste. 
The  following  works  are  contained  in  the  series :  — 

Moi,  par  Labiche  et  Martin. 

Gringoire,  par  Theodore  de  Banville,  et  L'Ete  de  la  Saint  Martin, 

par  Meiihac  et  Hel6vy. 
La  Question  d' Argent,  par  Alexandre  Dumas,//?. 
La  Camaraderie,  par  Eugene  Scribe. 

Le  Luthier  de  Cremone,  et  Le  Tr6sor,  par  Frangois  Copp6e. 
Le  Fils  de  Giboyer,  par  Emile  Augier. 

Professor  A.  G.  Cameron,  Yale  University :  The  volumes  are  as  admi- 
rable in  editing  as  they  are  dainty  in  form. 

Professor  H.  A.  Rennert,  University  of  Pennsylvania :  It  (Moi)  is  an 
excellent  book  in  every  way,  and  I  shall  use  it. 

Professor  George  D.  Fairfield,  University  of  Illinois  :  I  heartily  commend 
both  the  editorial  and  typographical  excellence  apparent  all  through 
it  (Moi). 

Professor  W.  A.  Cooper,  Marietta  College,  Ohio :  I  have  already  used 
two  of  the  Qa  Ira  Series  with  classes  and  shall  use  another  this  term. 
The  books  are  not  only  very  delightful  to  look  at,  but  the  editor's  work 
gives  the  student  just  what  help  he  needs. 


SCIENCE.  51 


Physics  for  Uniuersity  Students. 

By  Professor  Henry  S.  Carhart,  University  of  Michigan. 

Parti.     Mechanics,  Sound,  and  Light.     With  154  Illustrations.     i2mo, 

cloth,  330  pages.     Price,  ;^i.5o. 

Part  II.     Heat,  Electricity,  and  Magnetism.     With  224  Illustrations. 

i2mo,  cloth,  446  pages.     Price,  ^1.50. 

THESE  volumes,  the  outgrowth  of  long  experience  in  teach- 
ing, offer  a  full  course  in  University  Physics.  In  preparing 
the  work,  the  author  has  kept  constantly  in  view  the  actual  needs 
of  the  class-room.  The  result  is  a  fresh,  practical  text-book,  and 
not  a  cyclopaedia  of  physics. 

Particular  attention  has  been  given  to  the  arrangement  of 
topics,  so  as  to  secure  a  natural  and  logical  sequence.  In  many 
demonstrations  the  method  of  the  Calculus  is  used  without  its 
formal  symbols ;  and,  in  general,  mathematics  is  called  into  ser- 
vice, not  for  its  own  sake,  but  wholly  for  the  purpose  of  establish- 
ing the  relations  of  physical  quantities.  At  the  same  time  the 
course  in  Physics  represented  by  this  book  is  supposed  to  pre- 
cede the  study  of  calculus,  and  its  methods  will  in  a  general  way 
prepare  the  student  for  the  study  of  higher  mathematics. 

Professor  W.  LeConte  Stevens,  Rensselaer  Polytechnic  Institute,  Troy,  N.  Y.  : 
After  an  examination  of  Carhart's  University  Physics,  I  have  unhesitat- 
ingly decided  to  use  it  with  my  next  class.  The  book  is  admirably 
arranged,  clearly  expressed,  and  bears  the  unmistakable  mark  of  the 
work  of  a  successful  teacher. 

Professor  Florian  Cajori,  Colorado  College :  The  strong  features  of  his  Uni- 
versity Physics  appear  to  me  to  be  conciseness  and  accuracy  of  statement, 
the  emphasis  laid  on  the  more  important  topics  by  the  exclusion  of  minor 
details,  the  embodiment  of  recent  researches  whenever  possible. 

Professor  A.  A.  Atkinson,  Ohio  University,  Athens,  O. :  I  am  very  much 
pleased  with  the  book.  The  important  principles  of  physics  and  the 
essentials  of  energy  are  so  well  set  forth  for  the  student  for  which  the  book 
is  designed,  that  it  at  once  commends  itself  to  the  teacher. 

Professor  A.  E.  Frost,  Western  University,  Allegheny,  Pa. :  I  think  that  it 
comes  nearer  meeting  my  special  needs  than  any  book  I  have  examined, 
being  far  enough  above  the  High  School  book  to  justify  its  name,  and 
yet  not  so  far  above  it  as  to  be  a  discouragement  to  the  average  student. 


56  SCIENCE. 

Herbarium  and  Plant  Descriptions. 

Designed  by  Professor  EDWARD  T.  NELSON,  late  of  Ohio  Wesleyan 
University.    Portfolio,  7%  X  10  inches.     Price,  75  cents. 

THIS  is  an  herbarium  and  plant  record  combined,  enabling 
the  student  to  preserve  the  specimens  together  with  a 
record  of  their  characteristics. 

A  sheet  of  four  pages  is  devoted  to  each  specimen.  The  first 
page  contains  a  blank  form,  with  ample  space  for  a  full  descrip- 
tion of  the  plant,  and  for  notes  of  the  circumstances  under 
which  it  was  collected.  The  pressed  specimen  is  to  be  mounted 
on  the  third  page,  and  the  entire  sheet  then  serves  as  a  species- 
cover.  Each  portfolio  contains  fifty  sheets,  which  are  separate, 
so  as  to  permit  of  scientific  rearrangement  after  mounting  the 
specimens. 

The  preliminary  matter  gives  full  directions  for  collecting, 
pressing,  and  mounting  plants,  as  well  as  a  synopsis  of  botanical 
terms. 

The  portfolio  is  strong,  durable,  and  attractive  in  appearance. 

In  the  class-room  and  in  the  field  this  work  has  been  found 
helpful  and  stimulating.  It  encourages  observation  and  research, 
and  leads  to  an  exact  knowledge  of  classification. 

Professor  D.  P.  Penhallow,  McGill  University,  Montreal,  Can.:  The  idea 
is  a  good  one,  and  well  carried  out.  I  am  sure  it  will  prove  most  useful 
in  the  botanical  work  of  schools  and  academies,  for  which  I  would 
strongly  recommend  it. 

Professor  G.  H.  Perkins,  University  of  Vermont,  Burlington,  Vt.  :  It  is  the 
best  thing  of  the  sort  I  have  seen ;  very  attractive  and  very  helpful  to 
beginners  in  calling  attention  to  points  that  would  be  overlooked. 

Professor  B.  P.  Colton,  Normal  University,  III. :  It  is  a  very  ingenious  ar- 

-rangement,  and  neatly  gotten  up.     It  speaks  well  for  the  publishers,  as 

well  as  the  designer.     It  is  the  neatest  scheme  of  the  kind  I  have  seen. 

0.  D.  Robinson,  Principal  of  High  School,  Albany,  N.  Y. :  It  appears  to  me 
to  be  a  very  complete  arrangement,  admirable  in  every  respect,  and  very 
moderate  in  price. 

F.  S.  Hotaling,  Formerly  Principal  of  High  School,  Framingham,  Mass.  : 
Last  year's  work  in  botany  was  made  so  much  more  interesting  and  valua- 
ble by  the  use  of  the  Herbarium  that  we  find  it  now  a  necessity. 


SCIENCE.  hi 


Lessons  in  Elementary  Botany  for  Seco^idary 
Schools. 

By  Professor  Thomas  H.  MacBRIDE,  Iowa  State  University.     i6mo, 
cloth,  244  pages.     Price,  60  cents. 

THIS  book  is  designed  to  make  the  laboratory  study  of  botany 
possible  in  every  High  School  in  the  country.  While  call- 
ing for  the  use  of  none  but  the  simplest  appliances,  and  for  no 
material  but  such  as  is  easily  accessible  to  every  teacher,  its 
method  is  nevertheless  in  perfect  harmony  with  the  best  scien- 
tific instruction  of  the  time.  The  student  is  not  asked  to  study 
illustrations  or  text ;  he  is  sent  directly  to  the  plants  themselves, 
and  shown  how  to  study  these  and  observe  for  himself  the  vari- 
ous problems  of  vegetable  life.  The  plants  passed  in  review  are 
those  which  are  more  or  less  familiar  to  every  one,  and  the  life 
history  of  which  every  child  should  know. 

Austin  C.  Apgar,  State  Normal  School,  Trenton,  N.J.  :  There  are  many 
points  in  the  book  which  please  me.  Not  the  least  of  these  is  that  the 
author  has  not  been  misled  by  the  craze  for  that  "  logical  order  "  which 
begins  with  protoplasm  and,  some  time  or  other,  if  the  subject  be  pursued 
long  enough,  reaches  such  things  as  can  easily  be  found  and  examined. 
He  begins  with  the  known  and  gradually  advances  to  the  unknown, — 
the  only  order  in  which  successful  teacliing  can  be  accomplished. 

A  New  Flora. 

To  accompany  Macbride's  Elementary  Botany. 

IN  resjDonse  to  a  desire  expressed  by  many  teachers,  Professor 
Macbride  has  prepared  a  Key  to  the  More  Common  Species 
of  Native  and  Cultivated  Plants  occurring  in  the  Northern  United 
States.  The  material  thus  furnished  has  never  before  been 
offered  in  such  convenient  form  for  class-room  work.  Its  use  will 
in  no  way  change  the  original  plan  of  the  Elementary  Botany, 
but  it  offers  many  advantages  for  additional  study. 

The  Key  will  be  furnished  without  charge  to  those  ordering 
the  author's  Botany.  Copies  ordered  without  the  Botany  will  be 
furnished  at  twenty-five  cents  each. 


58  MA  THEM  A  TICS. 


Elements  of  Algebra. 

By  Professor  JAMES  M.  TAYLOR,  Colgate  University,  Hamilton,  N.Y. 
At  Press. 

IN  this  book  Professor  Taylor  aims  primarily  at  simplicity  in 
method  and  statement,  and  at  a  natural  and  logical  sequence 
in  the  series  of  steps  which  lead  the  pupil  from  his  arithmetic 
through  his  algebra. 

An  introductory  chapter  explains  the  meaning  and  object  of 
literal  notation,  and  illustrates  the  use  of  the  equation  in  solving 
arithmetical  problems.  This  is  followed  by  a  drill  on  particular 
numbers  before  the  pupil  is  introduced  to  the  use  of  letters  to 
represent  general  algebraic  numbers.  General  principles  are 
brought  out  by  induction  from  particular  cases,  and  proofs  are 
given  in  their  natural  places  where  the  pupil  will  be  unlikely  to 
memorize  without  comprehending  them.  Nomenclature  has  been 
looked  to  carefully.  Many  of  the  misleading  terms  of  the  older 
text-books  have  been  discarded  and  others  more  useful  and  help- 
ful have  been  applied. 

The  methods  of  working  examples  have  been  chosen  for  their 
simplicity  and  the  scope  of  their  application.  Suggestions  as  to 
method  of  attack  are  given,  but  formal  rules  are  stated  but  rarely. 
Positive  and  negative  numbers  are  so  explained  and  defined  as  to 
give  clear  and  true  concepts,  such  as  lead  naturally  to  still  broader 
views  of  numbers.  Factoring  is  made  a  fundamental  principle  in 
the  solution  of  quadratic  and  higher  equations.  Particular  atten- 
tion is  given  to  the  theory  of  equivalent  equations.  In  illustrating 
the  meaning  of  numbers,  equations,  and  systems  of  equations,  the 
graphic  method  is  used.  In  general  the  aim  is  to  render  as  clear 
as  possible  to  the  pupil  all  fundamental  processes  and  to  simplify 
the  statement  of  rules. 

The  book  is  particularly  adapted  to  beginners,  and  is  intended 
at  the  same  time  to  prepare  for  any  college  or  scientific  school, 
as  each  subject  is  so  treated  that  the  pupil  will  have  nothing  to 
unlearn  as  he  advances  in  mathematics. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below, 
or  on  the  date  to  which  renewed.  Renewals  only: 

Tel.  No.  642-3405 
Renewals  may  be  made  4  days  prior  to  date  due. 
Renewed  books  are  subject  to  immediate  recall. 


RY 


;low. 


>70    '  Jt 


Ni 


78 


LI 


LD21A-50m-2,'71 
(P2001810 )  476 — A-32 


General  Library 

University  of  California 

Berkeley 


P306047 


T3 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


